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Theorem bj-epelb 36253
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 5733 available. Check if it is shorter to prove bj-epelg 36252 first or bj-epelb 36253 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 5827 . . . 4 Rel E
21brrelex2i 5733 . . 3 (𝐴 E 𝐵𝐵 ∈ V)
32pm4.71i 560 . 2 (𝐴 E 𝐵 ↔ (𝐴 E 𝐵𝐵 ∈ V))
4 epelg 5581 . . 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 3474   class class class wbr 5148   E cep 5579
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-eprel 5580  df-xp 5682  df-rel 5683
This theorem is referenced by: (None)
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