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Theorem epelg 4950
 Description: The epsilon relation and membership are the same. General version of epel 4952. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
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 4584 . . . 4 (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2 elopab 4908 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦))
3 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
4 vex 3176 . . . . . . . . . . 11 𝑦 ∈ V
53, 4pm3.2i 470 . . . . . . . . . 10 (𝑥 ∈ V ∧ 𝑦 ∈ V)
6 opeqex 4887 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
75, 6mpbiri 247 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
87simpld 474 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
98adantr 480 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
109exlimivv 1847 . . . . . 6 (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
112, 10sylbi 206 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} → 𝐴 ∈ V)
12 df-eprel 4949 . . . . 5 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1311, 12eleq2s 2706 . . . 4 (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
141, 13sylbi 206 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
1514a1i 11 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
16 elex 3185 . . 3 (𝐴𝐵𝐴 ∈ V)
1716a1i 11 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
18 eleq12 2678 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
1918, 12brabga 4914 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
2019expcom 450 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
2115, 17, 20pm5.21ndd 368 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  {copab 4642   E cep 4947 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949 This theorem is referenced by:  epelc  4951  efrirr  5019  efrn2lp  5020  predep  5623  epne3  6872  cnfcomlem  8479  fpwwe2lem6  9336  ltpiord  9588  orvcelval  29857
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