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Theorem bj-epelg 36588
Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5589 and closed form of epeli 5588. (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 5738 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 5833 . . . 4 Rel E
21brrelex1i 5738 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
32a1i 11 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
4 elex 3492 . . 3 (𝐴𝐵𝐴 ∈ V)
54a1i 11 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
6 eleq12 2819 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
7 df-eprel 5586 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
86, 7brabga 5540 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
98expcom 412 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
103, 5, 9pm5.21ndd 378 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  Vcvv 3473   class class class wbr 5152   E cep 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-eprel 5586  df-xp 5688  df-rel 5689
This theorem is referenced by: (None)
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