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Theorem epelg 5465
Description: The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5468 and closed form of epeli 5467. (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 5064 . . . 4 (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2 0nelopab 5449 . . . . . . . 8 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
3 df-eprel 5464 . . . . . . . . . 10 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
43eqcomi 2835 . . . . . . . . 9 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} = E
54eleq2i 2909 . . . . . . . 8 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ↔ ∅ ∈ E )
62, 5mtbi 323 . . . . . . 7 ¬ ∅ ∈ E
7 eleq1 2905 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ E ↔ ∅ ∈ E ))
86, 7mtbiri 328 . . . . . 6 (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ E )
98con2i 141 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ E → ¬ ⟨𝐴, 𝐵⟩ = ∅)
10 opprc1 4826 . . . . 5 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
119, 10nsyl2 143 . . . 4 (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
121, 11sylbi 218 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
1312a1i 11 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
14 elex 3518 . . 3 (𝐴𝐵𝐴 ∈ V)
1514a1i 11 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
16 eleq12 2907 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
1716, 3brabga 5418 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
1817expcom 414 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
1913, 15, 18pm5.21ndd 381 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  Vcvv 3500  c0 4295  cop 4570   class class class wbr 5063  {copab 5125   E cep 5463
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-eprel 5464
This theorem is referenced by:  epeli  5467  efrirr  5535  efrn2lp  5536  predep  6173  epne3  7488  cnfcomlem  9156  fpwwe2lem6  10051  ltpiord  10303  orvcelval  31631  bj-epelb  34261  brcnvep  35413  epelon2  39771  alephiso2  39801
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