Theorem bj-ideqg 36764

Description: Characterization of the classes related by the identity relation when their intersection is a set. Note that the antecedent is more general than either class being a set. (Contributed by NM, 30-Apr-2004.) Weaken the antecedent to sethood of the intersection. (Revised by BJ, 24-Dec-2023.)

TODO: replace ideqg 5854, or at least prove ideqg 5854 from it.

Assertion

Ref	Expression
bj-ideqg	⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem bj-ideqg

Step	Hyp	Ref	Expression
1		df-br 5150	. 2 ⊢ (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I )
2		bj-opelid 36763	. 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
3	1, 2	bitrid 282	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098   ∩ cin 3943  ⟨cop 4636   class class class wbr 5149   I cid 5575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576

This theorem is referenced by:  bj-ideqb  36766