bookshelf/Bookshelf/Enderton/Set/OrderedPair.lean

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import Common.Set.Basic
2023-06-29 21:30:48 +00:00
/-! # Enderton.Set.OrderedPair
A representation of an ordered pair in basic set theory. Like `Set`, it is
assumed an ordered pair is homogeneous.
-/
/--
Kazimierz Kuratowski's definition of an ordered pair.
-/
def OrderedPair (x y : α) : Set (Set α) := {{x}, {x, y}}
namespace OrderedPair
/--
For any sets `x`, `y`, `u`, and `v`, `⟨u, v⟩ = ⟨x, y⟩` **iff** `u = x ∧ v = y`.
-/
theorem ext_iff {x y u v : α}
: (OrderedPair x y = OrderedPair u v) ↔ (x = u ∧ y = v) := by
unfold OrderedPair
apply Iff.intro
· intro h
have hu := Set.ext_iff.mp h {u}
have huv := Set.ext_iff.mp h {u, v}
simp only [
Set.mem_singleton_iff, Set.mem_insert_iff, true_or, or_true, iff_true
] at hu huv
apply Or.elim hu <;> apply Or.elim huv
· -- #### Case 1
-- `{u} = {x}` and `{u, v} = {x}`.
intro huv_x hu_x
rw [Set.singleton_eq_singleton_iff] at hu_x
rw [hu_x] at huv_x
have hx_v := Set.pair_eq_singleton_mem_imp_eq_self huv_x
rw [hu_x, hx_v] at h
simp only [Set.mem_singleton_iff, Set.insert_eq_of_mem] at h
have := Set.pair_eq_singleton_mem_imp_eq_self $
Set.pair_eq_singleton_mem_imp_eq_self h
rw [← hx_v] at this
exact ⟨hu_x.symm, this⟩
· -- #### Case 2
-- `{u} = {x}` and `{u, v} = {x, y}`.
intro huv_xy hu_x
rw [Set.singleton_eq_singleton_iff] at hu_x
rw [hu_x] at huv_xy
by_cases hx_v : x = v
· rw [hx_v] at huv_xy
simp only [Set.mem_singleton_iff, Set.insert_eq_of_mem] at huv_xy
have := Set.pair_eq_singleton_mem_imp_eq_self huv_xy.symm
exact ⟨hu_x.symm, this⟩
· rw [Set.ext_iff] at huv_xy
have := huv_xy v
simp only [
Set.mem_singleton_iff, Set.mem_insert_iff, or_true, true_iff
] at this
exact ⟨hu_x.symm, Or.elim this (absurd ·.symm hx_v) (·.symm)⟩
· -- #### Case 3
-- `{u} = {x, y}` and `{u, v} = {x}`.
intro huv_x hu_xy
rw [Set.ext_iff] at huv_x hu_xy
have hu_x := huv_x u
have hy_u := hu_xy y
simp only [
Set.mem_singleton_iff,
Set.mem_insert_iff,
true_or,
true_iff,
or_true,
iff_true
] at hu_x hy_u
apply Or.elim huv
· intro huv_x
rw [← hu_x, Set.ext_iff] at huv_x
have := huv_x v
simp only [
Set.mem_singleton_iff, Set.mem_insert_iff, or_true, true_iff
] at this
exact ⟨hu_x.symm, by rwa [this]⟩
· intro huv_xy
rw [hu_x, Set.ext_iff] at huv_xy
have := huv_xy v
simp only [
Set.mem_singleton_iff, Set.mem_insert_iff, or_true, true_iff
] at this
apply Or.elim this
· intro hv_x
rw [hy_u, hu_x]
exact ⟨rfl, hv_x.symm⟩
· intro hv_y
exact ⟨hu_x.symm, hv_y.symm⟩
· -- #### Case 4
-- `{u} = {x, y}` and `{u, v} = {x, y}`.
intro huv_xy hu_xy
rw [Set.ext_iff] at huv_xy hu_xy
have hx_u := hu_xy x
have hy_u := hu_xy y
simp only [
Set.mem_singleton_iff, Set.mem_insert_iff, true_or, iff_true, or_true
] at hx_u hy_u
have := huv_xy v
simp only [
Set.mem_singleton_iff, Set.mem_insert_iff, or_true, true_iff
] at this
apply Or.elim this
· intro hv_x
rw [hv_x, hx_u]
exact ⟨rfl, hy_u⟩
· intro hv_y
exact ⟨hx_u, hv_y.symm⟩
· intro h
rw [h.left, h.right]
end OrderedPair