Theorem bj-opelid 37179

Description: Characterization of the ordered pair elements of the identity relation when the intersection of their components are sets. Note that the antecedent is more general than either component being a set. (Contributed by BJ, 29-Mar-2020.)

Assertion

Ref	Expression
bj-opelid	⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))

Proof of Theorem bj-opelid

Step	Hyp	Ref	Expression
1		bj-inexeqex 37177	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	ex 412	. 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
3		bj-opelidb 37175	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
4		simpr 484	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
5		ancr 546	. . . 4 ⊢ ((𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴 = 𝐵 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)))
6	4, 5	impbid2 226	. . 3 ⊢ ((𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵))
7	3, 6	bitrid 283	. 2 ⊢ ((𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
8	2, 7	syl 17	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∩ cin 3930  ⟨cop 4612   I cid 5552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5187  df-id 5553

This theorem is referenced by:  bj-ideqg  37180  bj-opelidres  37184  bj-opelidb1ALT  37189  bj-elid4  37191