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Theorem bj-epelg 34975
Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5463 and closed form of epeli 5462. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) TODO: move it to the main section after reordering to have brrelex1i 5605 available. (Proof shortened by BJ, 14-Jul-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-epelg (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))

Proof of Theorem bj-epelg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rele 5697 . . . 4 Rel E
21brrelex1i 5605 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
32a1i 11 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
4 elex 3426 . . 3 (𝐴𝐵𝐴 ∈ V)
54a1i 11 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
6 eleq12 2827 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
7 df-eprel 5460 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
86, 7brabga 5415 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
98expcom 417 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
103, 5, 9pm5.21ndd 384 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2110  Vcvv 3408   class class class wbr 5053   E cep 5459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-eprel 5460  df-xp 5557  df-rel 5558
This theorem is referenced by: (None)
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