import Mathlib.Data.Set.Basic import Mathlib.Data.Set.Lattice import Bookshelf.Enderton.Set.Chapter_1 import Common.Logic.Basic import Common.Set.Basic /-! # Enderton.Chapter_2 Axioms and Operations -/ namespace Enderton.Set.Chapter_2 /-- ### Exercise 3.1 Assume that `A` is the set of integers divisible by `4`. Similarly assume that `B` and `C` are the sets of integers divisible by `9` and `10`, respectively. What is in `A ∩ B ∩ C`? -/ theorem exercise_3_1 {A B C : Set ℤ} (hA : A = { x | Dvd.dvd 4 x }) (hB : B = { x | Dvd.dvd 9 x }) (hC : C = { x | Dvd.dvd 10 x }) : ∀ x ∈ (A ∩ B ∩ C), (4 ∣ x) ∧ (9 ∣ x) ∧ (10 ∣ x) := by intro x h rw [Set.mem_inter_iff] at h have ⟨⟨ha, hb⟩, hc⟩ := h refine ⟨?_, ⟨?_, ?_⟩⟩ · rw [hA] at ha exact Set.mem_setOf.mp ha · rw [hB] at hb exact Set.mem_setOf.mp hb · rw [hC] at hc exact Set.mem_setOf.mp hc /-- ### Exercise 3.2 Give an example of sets `A` and `B` for which `⋃ A = ⋃ B` but `A ≠ B`. -/ theorem exercise_3_2 {A B : Set (Set ℕ)} (hA : A = {{1}, {2}}) (hB : B = {{1, 2}}) : Set.sUnion A = Set.sUnion B ∧ A ≠ B := by apply And.intro · show ⋃₀ A = ⋃₀ B ext x show (∃ t, t ∈ A ∧ x ∈ t) ↔ ∃ t, t ∈ B ∧ x ∈ t apply Iff.intro · intro ⟨t, ⟨ht, hx⟩⟩ rw [hA] at ht refine ⟨{1, 2}, ⟨by rw [hB]; simp, ?_⟩⟩ apply Or.elim ht <;> · intro ht' rw [ht'] at hx rw [hx] simp · intro ⟨t, ⟨ht, hx⟩⟩ rw [hB] at ht rw [ht] at hx apply Or.elim hx · intro hx' exact ⟨{1}, ⟨by rw [hA]; simp, by rw [hx']; simp⟩⟩ · intro hx' exact ⟨{2}, ⟨by rw [hA]; simp, by rw [hx']; simp⟩⟩ · show A ≠ B -- Find an element that exists in `B` but not in `A`. Extensionality -- concludes the proof. intro h rw [hA, hB, Set.ext_iff] at h have h₁ := h {1, 2} simp at h₁ rw [Set.ext_iff] at h₁ have h₂ := h₁ 2 simp at h₂ /-- ### Exercise 3.3 Show that every member of a set `A` is a subset of `U A`. (This was stated as an example in this section.) -/ theorem exercise_3_3 {A : Set (Set α)} : ∀ x ∈ A, x ⊆ ⋃₀ A := by intro x hx show ∀ y ∈ x, y ∈ { a | ∃ t, t ∈ A ∧ a ∈ t } intro y hy rw [Set.mem_setOf_eq] exact ⟨x, ⟨hx, hy⟩⟩ /-- ### Exercise 3.4 Show that if `A ⊆ B`, then `⋃ A ⊆ ⋃ B`. -/ theorem exercise_3_4 {A B : Set (Set α)} (h : A ⊆ B) : ⋃₀ A ⊆ ⋃₀ B := by show ∀ x ∈ { a | ∃ t, t ∈ A ∧ a ∈ t }, x ∈ { a | ∃ t, t ∈ B ∧ a ∈ t } intro x hx rw [Set.mem_setOf_eq] at hx have ⟨t, ⟨ht, hxt⟩⟩ := hx rw [Set.mem_setOf_eq] exact ⟨t, ⟨h ht, hxt⟩⟩ /-- ### Exercise 3.5 Assume that every member of `𝓐` is a subset of `B`. Show that `⋃ 𝓐 ⊆ B`. -/ theorem exercise_3_5 {𝓐 : Set (Set α)} (h : ∀ x ∈ 𝓐, x ⊆ B) : ⋃₀ 𝓐 ⊆ B := by show ∀ y ∈ { a | ∃ t, t ∈ 𝓐 ∧ a ∈ t }, y ∈ B intro y hy rw [Set.mem_setOf_eq] at hy have ⟨t, ⟨ht𝓐, hyt⟩⟩ := hy exact (h t ht𝓐) hyt /-- ### Exercise 3.6a Show that for any set `A`, `⋃ 𝓟 A = A`. -/ theorem exercise_3_6a : ⋃₀ (𝒫 A) = A := by show { a | ∃ t, t ∈ { t | t ⊆ A } ∧ a ∈ t } = A ext x apply Iff.intro · intro hx rw [Set.mem_setOf_eq] at hx have ⟨t, ⟨htl, htr⟩⟩ := hx rw [Set.mem_setOf_eq] at htl exact htl htr · intro hx rw [Set.mem_setOf_eq] exact ⟨A, ⟨by rw [Set.mem_setOf_eq], hx⟩⟩ /-- ### Exercise 3.6b Show that `A ⊆ 𝓟 ⋃ A`. Under what conditions does equality hold? -/ theorem exercise_3_6b : A ⊆ 𝒫 (⋃₀ A) ∧ (A = 𝒫 (⋃₀ A) ↔ ∃ B, A = 𝒫 B) := by apply And.intro · show ∀ x ∈ A, x ∈ { t | t ⊆ ⋃₀ A } intro x hx rw [Set.mem_setOf] exact exercise_3_3 x hx · apply Iff.intro · intro hA exact ⟨⋃₀ A, hA⟩ · intro ⟨B, hB⟩ conv => rhs; rw [hB, exercise_3_6a] exact hB /-- ### Exercise 3.7a Show that for any sets `A` and `B`, `𝓟 A ∩ 𝓟 B = 𝓟 (A ∩ B)`. -/ theorem exercise_3_7A : 𝒫 A ∩ 𝒫 B = 𝒫 (A ∩ B) := by suffices 𝒫 A ∩ 𝒫 B ⊆ 𝒫 (A ∩ B) ∧ 𝒫 (A ∩ B) ⊆ 𝒫 A ∩ 𝒫 B from subset_antisymm this.left this.right apply And.intro · unfold Set.powerset intro x hx simp only [Set.mem_inter_iff, Set.mem_setOf_eq] at hx rwa [Set.mem_setOf, Set.subset_inter_iff] · unfold Set.powerset simp intro x hA _ exact hA -- theorem false_of_false_iff_true : (false ↔ true) → false := by simp /-- ### Exercise 3.7b (i) Show that `𝓟 A ∪ 𝓟 B ⊆ 𝓟 (A ∪ B)`. -/ theorem exercise_3_7b_i : 𝒫 A ∪ 𝒫 B ⊆ 𝒫 (A ∪ B) := by unfold Set.powerset intro x hx simp at hx apply Or.elim hx · intro hA rw [Set.mem_setOf_eq] exact Set.subset_union_of_subset_left hA B · intro hB rw [Set.mem_setOf_eq] exact Set.subset_union_of_subset_right hB A /-- ### Exercise 3.7b (ii) Under what conditions does `𝓟 A ∪ 𝓟 B = 𝓟 (A ∪ B)`.? -/ theorem exercise_3_7b_ii : 𝒫 A ∪ 𝒫 B = 𝒫 (A ∪ B) ↔ A ⊆ B ∨ B ⊆ A := by unfold Set.powerset apply Iff.intro · intro h by_contra nh rw [not_or_de_morgan] at nh have ⟨a, hA⟩ := Set.not_subset.mp nh.left have ⟨b, hB⟩ := Set.not_subset.mp nh.right rw [Set.ext_iff] at h have hz := h {a, b} -- `hz` states that `{a, b} ⊆ A ∨ {a, b} ⊆ B ↔ {a, b} ⊆ A ∪ B`. We show the -- left-hand side is false but the right-hand side is true, yielding our -- contradiction. suffices ¬({a, b} ⊆ A ∨ {a, b} ⊆ B) by have hz₁ : a ∈ A ∪ B := by rw [Set.mem_union] exact Or.inl hA.left have hz₂ : b ∈ A ∪ B := by rw [Set.mem_union] exact Or.inr hB.left exact absurd (hz.mpr $ Set.mem_mem_imp_pair_subset hz₁ hz₂) this intro hAB exact Or.elim hAB (fun y => absurd (y $ show b ∈ {a, b} by simp) hB.right) (fun y => absurd (y $ show a ∈ {a, b} by simp) hA.right) · intro h ext x apply Or.elim h · intro hA apply Iff.intro · intro hx exact Or.elim hx (Set.subset_union_of_subset_left · B) (Set.subset_union_of_subset_right · A) · intro hx refine Or.inr (Set.Subset.trans hx ?_) exact subset_of_eq (Set.left_subset_union_eq_self hA) · intro hB apply Iff.intro · intro hx exact Or.elim hx (Set.subset_union_of_subset_left · B) (Set.subset_union_of_subset_right · A) · intro hx refine Or.inl (Set.Subset.trans hx ?_) exact subset_of_eq (Set.right_subset_union_eq_self hB) /-- ### Exercise 3.9 Give an example of sets `a` and `B` for which `a ∈ B` but `𝓟 a ∉ 𝓟 B`. -/ theorem exercise_3_9 (ha : a = {1}) (hB : B = {{1}}) : a ∈ B ∧ 𝒫 a ∉ 𝒫 B := by apply And.intro · rw [ha, hB] simp · intro h have h₁ : 𝒫 a = {∅, {1}} := by rw [ha] exact Set.powerset_singleton 1 have h₂ : 𝒫 B = {∅, {{1}}} := by rw [hB] exact Set.powerset_singleton {1} rw [h₁, h₂] at h simp at h apply Or.elim h · intro h rw [Set.ext_iff] at h have := h ∅ simp at this · intro h rw [Set.ext_iff] at h have := h 1 simp at this /-- ### Exercise 3.10 Show that if `a ∈ B`, then `𝓟 a ∈ 𝓟 𝓟 ⋃ B`. -/ theorem exercise_3_10 {B : Set (Set α)} (ha : a ∈ B) : 𝒫 a ∈ 𝒫 (𝒫 (⋃₀ B)) := by have h₁ := exercise_3_3 a ha have h₂ := Chapter_1.exercise_1_3 h₁ generalize hb : 𝒫 (⋃₀ B) = b conv => rhs; unfold Set.powerset rw [← hb, Set.mem_setOf_eq] exact h₂ /-- ### Exercise 4.11 (i) Show that for any sets `A` and `B`, `A = (A ∩ B) ∪ (A - B)`. -/ theorem exercise_4_11_i {A B : Set α} : A = (A ∩ B) ∪ (A \ B) := by show A = fun a => A a ∧ B a ∨ A a ∧ ¬B a suffices ∀ x, (A x ∧ (B x ∨ ¬B x)) = A x by conv => rhs; ext x; rw [← and_or_left, this] intro x refine propext ?_ apply Iff.intro · intro hx exact hx.left · intro hx exact ⟨hx, em (B x)⟩ /-- ### Exercise 4.11 (ii) Show that for any sets `A` and `B`, `A ∪ (B - A) = A ∪ B`. -/ theorem exercise_4_11_ii {A B : Set α} : A ∪ (B \ A) = A ∪ B := by show (fun a => A a ∨ B a ∧ ¬A a) = fun a => A a ∨ B a suffices ∀ x, ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x) by conv => lhs; ext x; rw [or_and_left, this x] intro x refine propext ?_ apply Iff.intro · intro hx exact hx.left · intro hx exact ⟨hx, em (A x)⟩ section variable {A B C : Set ℕ} variable {hA : A = {1, 2, 3}} variable {hB : B = {2, 3, 4}} variable {hC : C = {3, 4, 5}} lemma right_diff_eq_insert_one_three : A \ (B \ C) = {1, 3} := by rw [hA, hB, hC] ext x apply Iff.intro · intro hx unfold SDiff.sdiff Set.instSDiffSet Set.diff at hx unfold Membership.mem Set.instMembershipSet Set.Mem setOf at hx unfold insert Set.instInsertSet Set.insert setOf at hx have ⟨ha, hb⟩ := hx rw [not_and_de_morgan, not_or_de_morgan] at hb simp only [Set.mem_singleton_iff, not_not] at hb refine Or.elim ha Or.inl ?_ intro hy apply Or.elim hb · intro hz exact Or.elim hy (absurd · hz.left) Or.inr · intro hz refine Or.elim hz Or.inr ?_ intro hz₁ apply Or.elim hy <;> apply Or.elim hz₁ <;> · intro hz₂ hz₃ rw [hz₂] at hz₃ simp at hz₃ · intro hx unfold SDiff.sdiff Set.instSDiffSet Set.diff unfold Membership.mem Set.instMembershipSet Set.Mem setOf unfold insert Set.instInsertSet Set.insert setOf apply Or.elim hx · intro hy refine ⟨Or.inl hy, ?_⟩ intro hz rw [hy] at hz unfold Membership.mem Set.instMembershipSet Set.Mem at hz unfold singleton Set.instSingletonSet Set.singleton setOf at hz simp only at hz · intro hy refine ⟨Or.inr (Or.inr hy), ?_⟩ intro hz have hzr := hz.right rw [not_or_de_morgan] at hzr exact absurd hy hzr.left lemma left_diff_eq_singleton_one : (A \ B) \ C = {1} := by rw [hA, hB, hC] ext x apply Iff.intro · intro hx unfold SDiff.sdiff Set.instSDiffSet Set.diff at hx unfold Membership.mem Set.instMembershipSet Set.Mem setOf at hx unfold insert Set.instInsertSet Set.insert setOf at hx have ⟨⟨ha, hb⟩, hc⟩ := hx rw [not_or_de_morgan] at hb hc apply Or.elim ha · simp · intro hy apply Or.elim hy · intro hz exact absurd hz hb.left · intro hz exact absurd hz hc.left · intro hx refine ⟨⟨Or.inl hx, ?_⟩, ?_⟩ <;> · intro hy cases hy with | inl y => rw [hx] at y; simp at y | inr hz => cases hz with | inl y => rw [hx] at y; simp at y | inr y => rw [hx] at y; simp at y /-- ### Exercise 4.14 Show by example that for some sets `A`, `B`, and `C`, the set `A - (B - C)` is different from `(A - B) - C`. -/ theorem exercise_4_14 : A \ (B \ C) ≠ (A \ B) \ C := by rw [ @right_diff_eq_insert_one_three A B C hA hB hC, @left_diff_eq_singleton_one A B C hA hB hC ] intro h rw [Set.ext_iff] at h have := h 3 simp at this end /-- ### Exercise 4.16 Simplify: `[(A ∪ B ∪ C) ∩ (A ∪ B)] - [(A ∪ (B - C)) ∩ A]` -/ theorem exercise_4_16 {A B C : Set α} : ((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ (B \ C)) ∩ A) = B \ A := by calc ((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ (B \ C)) ∩ A) _ = (A ∪ B) \ ((A ∪ (B \ C)) ∩ A) := by rw [Set.union_inter_cancel_left] _ = (A ∪ B) \ A := by rw [Set.union_inter_cancel_left] _ = B \ A := by rw [Set.union_diff_left] /-! ### Exercise 4.17 Show that the following four conditions are equivalent. (a) `A ⊆ B` (b) `A - B = ∅` (c) `A ∪ B = B` (d) `A ∩ B = A` -/ theorem exercise_4_17_i {A B : Set α} (h : A ⊆ B) : A \ B = ∅ := by ext x apply Iff.intro · intro hx exact absurd (h hx.left) hx.right · intro hx exact False.elim hx theorem exercise_4_17_ii {A B : Set α} (h : A \ B = ∅) : A ∪ B = B := by suffices A ⊆ B from Set.left_subset_union_eq_self this show ∀ t, t ∈ A → t ∈ B intro t ht rw [Set.ext_iff] at h by_contra nt exact (h t).mp ⟨ht, nt⟩ theorem exercise_4_17_iii {A B : Set α} (h : A ∪ B = B) : A ∩ B = A := by suffices A ⊆ B from Set.inter_eq_left_iff_subset.mpr this exact Set.union_eq_right_iff_subset.mp h theorem exercise_4_17_iv {A B : Set α} (h : A ∩ B = A) : A ⊆ B := Set.inter_eq_left_iff_subset.mp h /-- ### Exercise 4.19 Is `𝒫 (A - B)` always equal to `𝒫 A - 𝒫 B`? Is it ever equal to `𝒫 A - 𝒫 B`? -/ theorem exercise_4_19 {A B : Set α} : 𝒫 (A \ B) ≠ (𝒫 A) \ (𝒫 B) := by intro h have he : ∅ ∈ 𝒫 (A \ B) := by simp have ne : ∅ ∉ (𝒫 A) \ (𝒫 B) := by simp rw [Set.ext_iff] at h have := h ∅ exact absurd (this.mp he) ne /-- ### Exercise 4.20 Let `A`, `B`, and `C` be sets such that `A ∪ B = A ∪ C` and `A ∩ B = A ∩ C`. Show that `B = C`. -/ theorem exercise_4_20 {A B C : Set α} (hu : A ∪ B = A ∪ C) (hi : A ∩ B = A ∩ C) : B = C := by ext x apply Iff.intro · intro hB by_cases hA : x ∈ A · have : x ∈ A ∩ B := Set.mem_inter hA hB rw [hi] at this exact this.right · have : x ∈ A ∪ B := Set.mem_union_right A hB rw [hu] at this exact Or.elim this (absurd · hA) (by simp) · intro hC by_cases hA : x ∈ A · have : x ∈ A ∩ C := Set.mem_inter hA hC rw [← hi] at this exact this.right · have : x ∈ A ∪ C := Set.mem_union_right A hC rw [← hu] at this exact Or.elim this (absurd · hA) (by simp) /-- ### Exercise 4.21 Show that `⋃ (A ∪ B) = (⋃ A) ∪ (⋃ B)`. -/ theorem exercise_4_21 {A B : Set (Set α)} : ⋃₀ (A ∪ B) = (⋃₀ A) ∪ (⋃₀ B) := by ext x apply Iff.intro · intro hx have ⟨t, ht⟩ : ∃ t, t ∈ A ∪ B ∧ x ∈ t := hx apply Or.elim ht.left · intro hA exact Or.inl ⟨t, ⟨hA, ht.right⟩⟩ · intro hB exact Or.inr ⟨t, ⟨hB, ht.right⟩⟩ · intro hx apply Or.elim hx · intro hA have ⟨t, ht⟩ : ∃ t, t ∈ A ∧ x ∈ t := hA exact ⟨t, ⟨Set.mem_union_left B ht.left, ht.right⟩⟩ · intro hB have ⟨t, ht⟩ : ∃ t, t ∈ B ∧ x ∈ t := hB exact ⟨t, ⟨Set.mem_union_right A ht.left, ht.right⟩⟩ /-- ### Exercise 4.22 Show that if `A` and `B` are nonempty sets, then `⋂ (A ∪ B) = ⋂ A ∩ ⋂ B`. -/ theorem exercise_4_22 {A B : Set (Set α)} : ⋂₀ (A ∪ B) = ⋂₀ A ∩ ⋂₀ B := by ext x apply Iff.intro · intro hx have : ∀ t : Set α, t ∈ A ∪ B → x ∈ t := hx show (∀ t : Set α, t ∈ A → x ∈ t) ∧ (∀ t : Set α, t ∈ B → x ∈ t) rw [← forall_and] intro t exact ⟨ fun ht => this t (Set.mem_union_left B ht), fun ht => this t (Set.mem_union_right A ht) ⟩ · intro hx have : ∀ t : Set α, (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t) := by have : (∀ t : Set α, t ∈ A → x ∈ t) ∧ (∀ t : Set α, t ∈ B → x ∈ t) := hx rwa [← forall_and] at this show ∀ (t : Set α), t ∈ A ∪ B → x ∈ t intro t ht apply Or.elim ht · intro hA exact (this t).left hA · intro hB exact (this t).right hB /-- ### Exercise 4.24a Show that is `𝓐` is nonempty, then `𝒫 (⋂ 𝓐) = ⋂ { 𝒫 X | X ∈ 𝓐 }`. -/ theorem exercise_4_24a {𝓐 : Set (Set α)} : 𝒫 (⋂₀ 𝓐) = ⋂₀ { 𝒫 X | X ∈ 𝓐 } := by calc 𝒫 (⋂₀ 𝓐) _ = { x | x ⊆ ⋂₀ 𝓐 } := rfl _ = { x | x ⊆ { y | ∀ X, X ∈ 𝓐 → y ∈ X } } := rfl _ = { x | ∀ t ∈ x, t ∈ { y | ∀ X, X ∈ 𝓐 → y ∈ X } } := rfl _ = { x | ∀ t ∈ x, (∀ X, X ∈ 𝓐 → t ∈ X) } := rfl _ = { x | ∀ X ∈ 𝓐, (∀ t, t ∈ x → t ∈ X) } := by ext rw [Set.mem_setOf, Set.mem_setOf, forall_mem_comm (· ∈ ·)] _ = { x | ∀ X ∈ 𝓐, x ⊆ X} := rfl _ = { x | ∀ X ∈ 𝓐, x ∈ 𝒫 X } := rfl _ = { x | ∀ t ∈ { 𝒫 X | X ∈ 𝓐 }, x ∈ t} := by simp _ = ⋂₀ { 𝒫 X | X ∈ 𝓐 } := rfl /-- ### Exercise 4.24b Show that ``` ⋃ {𝒫 X | X ∈ 𝓐} ⊆ 𝒫 ⋃ 𝓐. ``` Under what conditions does equality hold? -/ theorem exercise_4_24b {𝓐 : Set (Set α)} : (⋃₀ { 𝒫 X | X ∈ 𝓐 } ⊆ 𝒫 ⋃₀ 𝓐) ∧ ((⋃₀ { 𝒫 X | X ∈ 𝓐 } = 𝒫 ⋃₀ 𝓐) ↔ (⋃₀ 𝓐 ∈ 𝓐)) := by have hS : (⋃₀ { 𝒫 X | X ∈ 𝓐 } ⊆ 𝒫 ⋃₀ 𝓐) := by simp exact exercise_3_3 refine ⟨hS, ?_⟩ apply Iff.intro · intro rS have rS : 𝒫 ⋃₀ 𝓐 ⊆ ⋃₀ { 𝒫 X | X ∈ 𝓐 } := (Set.Subset.antisymm_iff.mp rS).right have hA : ⋃₀ 𝓐 ∈ ⋃₀ { 𝒫 X | X ∈ 𝓐 } := rS Set.self_mem_powerset_self conv at hA => rhs unfold Set.sUnion sSup Set.instSupSetSet simp only have ⟨X, ⟨⟨x, hx⟩, ht⟩⟩ := Set.mem_setOf.mp hA have : ⋃₀ 𝓐 = x := by rw [← hx.right] at ht have hl : ⋃₀ 𝓐 ⊆ x := ht have hr : x ⊆ ⋃₀ 𝓐 := exercise_3_3 x hx.left exact Set.Subset.antisymm hl hr rw [← this] at hx exact hx.left · intro hA suffices 𝒫 ⋃₀ 𝓐 ⊆ ⋃₀ { 𝒫 X | X ∈ 𝓐 } from subset_antisymm hS this show ∀ x, x ∈ 𝒫 ⋃₀ 𝓐 → x ∈ ⋃₀ { x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x } intro x hx unfold Set.sUnion sSup Set.instSupSetSet simp only [Set.mem_setOf_eq, exists_exists_and_eq_and, Set.mem_powerset_iff] exact ⟨⋃₀ 𝓐, ⟨hA, hx⟩⟩ /-- ### Exercise 4.25 Is `A ∪ (⋃ 𝓑)` always the same as `⋃ { A ∪ X | X ∈ 𝓑 }`? If not, then under what conditions does equality hold? -/ theorem exercise_4_25 {A : Set α} (𝓑 : Set (Set α)) : (A ∪ (⋃₀ 𝓑) = ⋃₀ { A ∪ X | X ∈ 𝓑 }) ↔ (A = ∅ ∨ Set.Nonempty 𝓑) := by apply Iff.intro · intro h by_cases h𝓑 : Set.Nonempty 𝓑 · exact Or.inr h𝓑 · have : 𝓑 = ∅ := Set.not_nonempty_iff_eq_empty.mp h𝓑 rw [this] at h simp at h exact Or.inl h · intro h apply Or.elim h · intro hA rw [hA] simp · intro h𝓑 calc A ∪ (⋃₀ 𝓑) _ = { x | x ∈ A ∨ x ∈ ⋃₀ 𝓑} := rfl _ = { x | x ∈ A ∨ (∃ b ∈ 𝓑, x ∈ b) } := rfl _ = { x | ∃ b ∈ 𝓑, x ∈ A ∨ x ∈ b } := by ext x rw [Set.mem_setOf, Set.mem_setOf] apply Iff.intro · intro hx apply Or.elim hx · intro hA have ⟨b, hb⟩ := Set.nonempty_def.mp h𝓑 exact ⟨b, ⟨hb, Or.inl hA⟩⟩ · intro ⟨b, hb⟩ exact ⟨b, ⟨hb.left, Or.inr hb.right⟩⟩ · intro ⟨b, ⟨hb, hx⟩⟩ apply Or.elim hx · exact (Or.inl ·) · intro h exact Or.inr ⟨b, ⟨hb, h⟩⟩ _ = { x | ∃ b ∈ 𝓑, x ∈ A ∪ b } := rfl _ = { x | ∃ t, t ∈ { y | ∃ X, X ∈ 𝓑 ∧ A ∪ X = y } ∧ x ∈ t } := by simp _ = ⋃₀ { A ∪ X | X ∈ 𝓑 } := rfl end Enderton.Set.Chapter_2