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Theorem epelg 5476
Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5478 and closed form of epeli 5477. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 14-Jul-2023.)
Assertion
Ref Expression
epelg (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))

Proof of Theorem epelg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5069 . . . 4 (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2 0nelopab 5461 . . . . . . . 8 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
3 df-eprel 5475 . . . . . . . . . 10 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
43eqcomi 2747 . . . . . . . . 9 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} = E
54eleq2i 2830 . . . . . . . 8 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ↔ ∅ ∈ E )
62, 5mtbi 325 . . . . . . 7 ¬ ∅ ∈ E
7 eleq1 2826 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ E ↔ ∅ ∈ E ))
86, 7mtbiri 330 . . . . . 6 (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ E )
98con2i 141 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ E → ¬ ⟨𝐴, 𝐵⟩ = ∅)
10 opprc1 4823 . . . . 5 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
119, 10nsyl2 143 . . . 4 (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
121, 11sylbi 220 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
1312a1i 11 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
14 elex 3439 . . 3 (𝐴𝐵𝐴 ∈ V)
1514a1i 11 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
16 eleq12 2828 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
1716, 3brabga 5430 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
1817expcom 417 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
1913, 15, 18pm5.21ndd 384 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2111  Vcvv 3421  c0 4252  cop 4562   class class class wbr 5068  {copab 5130   E cep 5474
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 2113  ax-9 2121  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
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 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2942  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-br 5069  df-opab 5131  df-eprel 5475
This theorem is referenced by:  epeli  5477  efrirr  5547  efrn2lp  5548  epin  5978  predep  6206  epne3  7577  cnfcomlem  9339  fpwwe2lem5  10274  ltpiord  10526  orvcelval  32174  bj-epelb  35005  brcnvep  36171  epelon2  40846  alephiso2  40874
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