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Theorem bj-ideqg 37159
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 5861, or at least prove ideqg 5861 from it.

Assertion
Ref Expression
bj-ideqg ((𝐴𝐵) ∈ 𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Proof of Theorem bj-ideqg
StepHypRef Expression
1 df-br 5143 . 2 (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I )
2 bj-opelid 37158 . 2 ((𝐴𝐵) ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
31, 2bitrid 283 1 ((𝐴𝐵) ∈ 𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  cin 3949  cop 4631   class class class wbr 5142   I cid 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577
This theorem is referenced by:  bj-ideqb  37161
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