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Theorem bj-epelg 37069
Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5587 and closed form of epeli 5586. (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 5741 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 5837 . . . 4 Rel E
21brrelex1i 5741 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
32a1i 11 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
4 elex 3501 . . 3 (𝐴𝐵𝐴 ∈ V)
54a1i 11 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
6 eleq12 2831 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
7 df-eprel 5584 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
86, 7brabga 5539 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
98expcom 413 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
103, 5, 9pm5.21ndd 379 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  Vcvv 3480   class class class wbr 5143   E cep 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-rel 5692
This theorem is referenced by: (None)
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