Enderton. Exercise 3.19.
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@ -4227,7 +4227,7 @@ Evaluate the following: $R \circ R$, $R \restriction \{1\}$,
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\end{proof}
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\subsection{\pending{Exercise 3.19}}%
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\subsection{\verified{Exercise 3.19}}%
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\label{sub:exercise-3.19}
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Let $$A = \{
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@ -4242,6 +4242,38 @@ Evaluate each of the following: $A(\emptyset)$, $\img{A}{\emptyset}$,
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\begin{proof}
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\statementpadding
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_i}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_ii}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_iii}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_iv}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_v}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_vi}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_vii}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_viii}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_ix}
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\lean{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_3\_19\_x}
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\begin{enumerate}[(i)]
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\item $A(\emptyset) = \{\emptyset, \{\emptyset\}\}$.
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\item $\img{A}{\emptyset} = \emptyset$.
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@ -923,7 +923,7 @@ Evaluate the following: `R ∘ R`, `R ↾ {1}`, `R⁻¹ ↾ {1}`, `R⟦{1}⟧`,
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`R⁻¹⟦{1}⟧`.
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-/
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section
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section Exercise_3_18
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variable {R : Set.Relation ℕ}
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variable (hR : R = {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)})
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@ -1044,7 +1044,257 @@ theorem exercise_3_18_v
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ext y
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simp
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end
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end Exercise_3_18
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/-! #### Exercise 3.19
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Let
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```
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A = {⟨∅, {∅, {∅}}⟩, ⟨{∅}, ∅⟩}.
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```
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Evaluate each of the following: `A(∅)`, `A⟦∅⟧`, `A⟦{∅}⟧`, `A⟦{∅, {∅}}⟧`,
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`A⁻¹`, `A ∘ A`, `A ↾ ∅`, `A ↾ {∅, {∅}}`, `⋃ ⋃ A`.
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-/
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section Exercise_3_19
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variable {A : Set.Relation (Set (Set (Set α)))}
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variable (hA : A = {(∅, {∅, {∅}}), ({∅}, ∅)})
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theorem exercise_3_19_i
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: (∅, {∅, {∅}}) ∈ A := by
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rw [hA]
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simp
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theorem exercise_3_19_ii
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: A.image ∅ = ∅ := by
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unfold image
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simp
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theorem exercise_3_19_iii
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: A.image {∅} = {{∅, {∅}}} := by
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unfold image
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rw [hA]
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ext x
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simp only [
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Set.mem_singleton_iff,
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Prod.mk.injEq,
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Set.mem_insert_iff,
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exists_eq_left,
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true_and
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]
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apply Iff.intro
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· intro hx
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simp at hx
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apply Or.elim hx
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· simp
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· intro ⟨h, _⟩
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exfalso
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rw [Set.ext_iff] at h
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have := h ∅
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simp at this
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· intro hx
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rw [hx]
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simp
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theorem exercise_3_19_iv
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: A.image {∅, {∅}} = {{∅, {∅}}, ∅} := by
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unfold image
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rw [hA]
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ext x
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simp only [
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Set.mem_singleton_iff,
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Set.mem_insert_iff,
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Prod.mk.injEq,
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exists_eq_or_imp,
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true_and,
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exists_eq_left,
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Set.singleton_ne_empty,
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false_and,
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false_or,
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Set.mem_setOf_eq
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]
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apply Iff.intro
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· intro h
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apply Or.elim h
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· intro hx₁
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apply Or.elim hx₁
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· intro hx₂; left ; exact hx₂
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· intro hx₂; right; exact hx₂.right
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· intro hx₂
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right
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exact hx₂
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· intro h
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apply Or.elim h
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· intro hx₁; iterate 2 left
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exact hx₁
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· intro hx₁; right; exact hx₁
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theorem exercise_3_19_v
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: A.inv = {({∅, {∅}}, ∅), (∅, {∅})} := by
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unfold inv
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rw [hA]
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ext x
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simp only [
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Set.mem_singleton_iff,
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Prod.mk.injEq,
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Set.mem_insert_iff,
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exists_eq_or_imp,
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exists_eq_left,
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Set.mem_setOf_eq
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]
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apply Iff.intro
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· intro hx
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apply Or.elim hx
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· intro hx₁; left ; rw [← hx₁]
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· intro hx₁; right; rw [← hx₁]
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· intro hx
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apply Or.elim hx
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· intro hx₁; left ; rw [← hx₁]
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· intro hx₁; right; rw [← hx₁]
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theorem exercise_3_19_vi
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: A.comp A = {({∅}, {∅, {∅}})} := by
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unfold comp
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rw [hA]
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ext x
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have (a, b) := x
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simp only [
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Set.mem_singleton_iff, Prod.mk.injEq, Set.mem_insert_iff, Set.mem_setOf_eq
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]
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apply Iff.intro
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· intro ⟨t, ht₁, ht₂⟩
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casesm* _ ∨ _
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all_goals case _ hl hr => first
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| {
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rw [hl.right] at hr
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have := hr.left
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rw [Set.ext_iff] at this
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simp at this
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}
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| exact ⟨hl.left, hr.right⟩
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· intro ⟨ha, hb⟩
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refine ⟨∅, ?_, ?_⟩
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· right; rw [ha]; simp
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· left ; rw [hb]; simp
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theorem exercise_3_19_vii
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: A.restriction ∅ = ∅ := by
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unfold restriction
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rw [hA]
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simp
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theorem exercise_3_19_viii
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: A.restriction {∅} = {(∅, {∅, {∅}})} := by
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unfold restriction
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rw [hA]
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ext x
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have (a, b) := x
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simp only [
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Set.mem_singleton_iff, Prod.mk.injEq, Set.mem_insert_iff, Set.mem_setOf_eq
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]
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apply Iff.intro
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· intro ⟨h, ha⟩
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apply Or.elim h
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· simp
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· intro ⟨ha', _⟩
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exfalso
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rw [ha', Set.ext_iff] at ha
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simp at ha
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· intro ⟨ha, hb⟩
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rw [ha, hb]
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simp
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theorem exercise_3_19_ix
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: A.restriction {∅, {∅}} = A := by
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unfold restriction
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rw [hA]
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ext x
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have (a, b) := x
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simp only [
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Set.mem_singleton_iff,
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Prod.mk.injEq,
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Set.mem_insert_iff,
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Set.mem_setOf_eq
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]
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apply Iff.intro
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· intro ⟨h₁, h₂⟩
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casesm* _ ∨ _
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· case _ hl _ => left ; exact hl
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· case _ hl _ => left ; exact hl
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· case _ hl _ => right; exact hl
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· case _ hl _ => right; exact hl
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· intro h₁
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apply Or.elim h₁ <;>
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· intro ⟨ha, hb⟩
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rw [ha, hb]
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simp
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theorem exercise_3_19_x
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: ⋃₀ ⋃₀ A.toOrderedPairs = {∅, {∅}, {∅, {∅}}} := by
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unfold toOrderedPairs OrderedPair Set.sUnion sSup Set.instSupSetSet
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rw [hA]
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ext x
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simp only [
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Set.mem_singleton_iff,
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Prod.mk.injEq,
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Set.mem_image,
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Set.mem_insert_iff,
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exists_eq_or_imp,
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exists_eq_left,
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Set.singleton_ne_empty,
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Set.mem_setOf_eq
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]
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apply Iff.intro
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· intro ⟨a, ⟨t, ht₁, ht₂⟩, hx⟩
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apply Or.elim ht₁
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· intro ht
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rw [← ht] at ht₂
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simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at ht₂
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apply Or.elim ht₂
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· intro ha
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rw [ha] at hx
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simp only [Set.mem_singleton_iff] at hx
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left
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exact hx
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· intro ha
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rw [ha] at hx
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simp at hx
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apply Or.elim hx <;>
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· intro hx'; rw [hx']; simp
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· intro ht
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rw [← ht] at ht₂
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simp only [
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Set.mem_singleton_iff, Set.singleton_ne_empty, Set.mem_insert_iff
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] at ht₂
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apply Or.elim ht₂
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· intro ha
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rw [ha] at hx
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simp only [Set.mem_singleton_iff] at hx
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rw [hx]
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simp
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· intro ha
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rw [ha] at hx
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simp only [
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Set.mem_singleton_iff, Set.singleton_ne_empty, Set.mem_insert_iff
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] at hx
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apply Or.elim hx <;>
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· intro hx'; rw [hx']; simp
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· intro hx
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apply Or.elim hx
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· intro hx₁
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rw [hx₁]
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refine ⟨{{∅}, ∅}, ⟨{{{∅}}, {{∅}, ∅}}, ?_⟩, ?_⟩ <;> simp
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· intro hx₁
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apply Or.elim hx₁
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· intro hx₂
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rw [hx₂]
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refine ⟨{{∅}, ∅}, ⟨{{{∅}}, {{∅}, ∅}}, ?_⟩, ?_⟩ <;> simp
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· intro hx₂
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rw [hx₂]
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refine ⟨{∅, {∅, {∅}}}, ⟨{{∅}, {∅, {∅, {∅}}}}, ?_⟩, ?_⟩ <;> simp
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end Exercise_3_19
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end Relation
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