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Theorem bj-epelb 35471
Description: Two classes are related by the membership relation if and only if they are related by the membership relation (i.e., the first is an element of the second) and the second is a set (hence so is the first). TODO: move to Main after reordering to have brrelex2i 5687 available. Check if it is shorter to prove bj-epelg 35470 first or bj-epelb 35471 first. (Contributed by BJ, 14-Jul-2023.)
Assertion
Ref Expression
bj-epelb (𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))

Proof of Theorem bj-epelb
StepHypRef Expression
1 rele 5781 . . . 4 Rel E
21brrelex2i 5687 . . 3 (𝐴 E 𝐵𝐵 ∈ V)
32pm4.71i 560 . 2 (𝐴 E 𝐵 ↔ (𝐴 E 𝐵𝐵 ∈ V))
4 epelg 5536 . . 3 (𝐵 ∈ V → (𝐴 E 𝐵𝐴𝐵))
54pm5.32ri 576 . 2 ((𝐴 E 𝐵𝐵 ∈ V) ↔ (𝐴𝐵𝐵 ∈ V))
63, 5bitri 274 1 (𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wcel 2106  Vcvv 3443   class class class wbr 5103   E cep 5534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-eprel 5535  df-xp 5637  df-rel 5638
This theorem is referenced by: (None)
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