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Theorem bj-ideqg 37140
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 5865, or at least prove ideqg 5865 from it.

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

Proof of Theorem bj-ideqg
StepHypRef Expression
1 df-br 5149 . 2 (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I )
2 bj-opelid 37139 . 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 1537  wcel 2106  cin 3962  cop 4637   class class class wbr 5148   I cid 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583
This theorem is referenced by:  bj-ideqb  37142
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