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Theorem bj-epelb 34377
 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 5582 available. Check if it is shorter to prove bj-epelg 34376 first or bj-epelb 34377 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 5672 . . . 4 Rel E
21brrelex2i 5582 . . 3 (𝐴 E 𝐵𝐵 ∈ V)
32pm4.71i 563 . 2 (𝐴 E 𝐵 ↔ (𝐴 E 𝐵𝐵 ∈ V))
4 epelg 5439 . . 3 (𝐵 ∈ V → (𝐴 E 𝐵𝐴𝐵))
54pm5.32ri 579 . 2 ((𝐴 E 𝐵𝐵 ∈ V) ↔ (𝐴𝐵𝐵 ∈ V))
63, 5bitri 278 1 (𝐴 E 𝐵 ↔ (𝐴𝐵𝐵 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2115  Vcvv 3471   class class class wbr 5039   E cep 5437 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-eprel 5438  df-xp 5534  df-rel 5535 This theorem is referenced by: (None)
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