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Theorem bj-epelg 34635
 Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5437 and closed form of epeli 5436. (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 5576 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 5667 . . . 4 Rel E
21brrelex1i 5576 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
32a1i 11 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
4 elex 3460 . . 3 (𝐴𝐵𝐴 ∈ V)
54a1i 11 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
6 eleq12 2879 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
7 df-eprel 5434 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
86, 7brabga 5390 . . 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 2111  Vcvv 3442   class class class wbr 5034   E cep 5433 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-eprel 5434  df-xp 5529  df-rel 5530 This theorem is referenced by: (None)
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