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Theorem bj-ideqb 37160
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 5836 . . 3 Rel I
21brrelex1i 5741 . 2 (𝐴 I 𝐵𝐴 ∈ V)
3 inex1g 5319 . . 3 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
4 bj-ideqg 37158 . . 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 1540  wcel 2108  Vcvv 3480  cin 3950   class class class wbr 5143   I cid 5577
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692
This theorem is referenced by: (None)
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