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Theorem epelg 5585
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. Definition 1.6 of [Schloeder] p. 1. (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 5144 . . . 4 (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2 0nelopab 5572 . . . . . . . 8 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
3 df-eprel 5584 . . . . . . . . . 10 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
43eqcomi 2746 . . . . . . . . 9 {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} = E
54eleq2i 2833 . . . . . . . 8 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ↔ ∅ ∈ E )
62, 5mtbi 322 . . . . . . 7 ¬ ∅ ∈ E
7 eleq1 2829 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ∅ → (⟨𝐴, 𝐵⟩ ∈ E ↔ ∅ ∈ E ))
86, 7mtbiri 327 . . . . . 6 (⟨𝐴, 𝐵⟩ = ∅ → ¬ ⟨𝐴, 𝐵⟩ ∈ E )
98con2i 139 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ E → ¬ ⟨𝐴, 𝐵⟩ = ∅)
10 opprc1 4897 . . . . 5 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
119, 10nsyl2 141 . . . 4 (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
121, 11sylbi 217 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
1312a1i 11 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
14 elex 3501 . . 3 (𝐴𝐵𝐴 ∈ V)
1514a1i 11 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
16 eleq12 2831 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
1716, 3brabga 5539 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
1817expcom 413 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
1913, 15, 18pm5.21ndd 379 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cop 4632   class class class wbr 5143  {copab 5205   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-ne 2941  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
This theorem is referenced by:  epeli  5586  efrirr  5665  efrn2lp  5666  epin  6113  predep  6351  epne3  7793  cnfcomlem  9739  fpwwe2lem5  10675  ltpiord  10927  orvcelval  34471  bj-epelb  37070  brcnvep  38266  onsupuni  43241  oninfint  43248  onepsuc  43264  cantnfresb  43337  epelon2  43534  alephiso2  43571
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