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Theorem bj-epelb 35353
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 5675 available. Check if it is shorter to prove bj-epelg 35352 first or bj-epelb 35353 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 5769 . . . 4 Rel E
21brrelex2i 5675 . . 3 (𝐴 E 𝐵𝐵 ∈ V)
32pm4.71i 560 . 2 (𝐴 E 𝐵 ↔ (𝐴 E 𝐵𝐵 ∈ V))
4 epelg 5525 . . 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 2105  Vcvv 3441   class class class wbr 5092   E cep 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-eprel 5524  df-xp 5626  df-rel 5627
This theorem is referenced by: (None)
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