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Theorem bj-ideqb 37142
Description: Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
Assertion
Ref Expression
bj-ideqb (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-ideqb
StepHypRef Expression
1 reli 5839 . . 3 Rel I
21brrelex1i 5745 . 2 (𝐴 I 𝐵𝐴 ∈ V)
3 inex1g 5325 . . 3 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
4 bj-ideqg 37140 . . 3 ((𝐴𝐵) ∈ V → (𝐴 I 𝐵𝐴 = 𝐵))
53, 4syl 17 . 2 (𝐴 ∈ V → (𝐴 I 𝐵𝐴 = 𝐵))
62, 5biadanii 822 1 (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962   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-ral 3060  df-rex 3069  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  df-xp 5695  df-rel 5696
This theorem is referenced by: (None)
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