bookshelf/Common/Set/OrderedPair.lean

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2023-06-07 02:16:06 +00:00
import Mathlib.Data.Set.Basic
import Common.Logic.Basic
import Common.Set.Basic
namespace Set
/--
Kazimierz Kuratowski's definition of an ordered pair.
-/
def OrderedPair (x y : α) : Set (Set α) := {{x}, {x, y}}
namespace OrderedPair
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 h' := h
rw [Set.ext_iff] at h'
have hu := h' {u}
have huv := h' {u, v}
simp only [mem_singleton_iff, mem_insert_iff, true_or, iff_true] at hu
simp only [mem_singleton_iff, mem_insert_iff, or_true, iff_true] at huv
apply Or.elim hu
· apply Or.elim huv
· -- #### Case 1
-- `{u} = {x}` and `{u, v} = {x}`.
intro huv_x hu_x
rw [singleton_eq_singleton_iff] at hu_x
rw [hu_x] at huv_x
have hx_v := pair_eq_singleton_mem_imp_eq_self huv_x
rw [hu_x, hx_v] at h
simp only [mem_singleton_iff, insert_eq_of_mem] at h
have := pair_eq_singleton_mem_imp_eq_self $
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 [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 at huv_xy
have := 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 at this
apply Or.elim this
· intro hv_x
rw [hu_x, ← hv_x] at h
simp at h
have := pair_eq_singleton_mem_imp_eq_self $
pair_eq_singleton_mem_imp_eq_self h
exact ⟨hu_x.symm, this⟩
· intro hv_y
exact ⟨hu_x.symm, hv_y.symm⟩
· apply Or.elim huv
· -- #### Case 3
-- `{u} = {x, y}` and `{u, v} = {x}`.
intro huv_x hu_xy
rw [Set.ext_iff] at huv_x
have hu_x := huv_x u
have hv_x := huv_x v
simp only [mem_singleton_iff, mem_insert_iff, true_or, true_iff] at hu_x
simp only [mem_singleton_iff, mem_insert_iff, or_true, true_iff] at hv_x
rw [← hu_x] at hu_xy
have := pair_eq_singleton_mem_imp_eq_self hu_xy.symm
rw [hu_x, ← hv_x] at this
exact ⟨hu_x.symm, this⟩
· -- #### Case 4
-- `{u} = {x, y}` and `{u, v} = {x, y}`.
intro huv_xy hu_xy
rw [Set.ext_iff] at hu_xy
have hx_u := hu_xy x
have hy_u := hu_xy y
simp only [mem_singleton_iff, mem_insert_iff, true_or, iff_true] at hx_u
simp only [mem_singleton_iff, mem_insert_iff, or_true, iff_true] at hy_u
rw [hx_u, hy_u] at huv_xy
simp only [mem_singleton_iff, insert_eq_of_mem] at huv_xy
have := pair_eq_singleton_mem_imp_eq_self huv_xy
rw [← this] at hy_u
exact ⟨hx_u, hy_u⟩
· intro h
rw [h.left, h.right]
end OrderedPair
end Set