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Theorem epelg 4108
Description: The epsilon relation and membership are the same. General version of epel 4110. (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 3838 . . . 4 (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2 elopab 4076 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦))
3 vex 2622 . . . . . . . . . . 11 𝑥 ∈ V
4 vex 2622 . . . . . . . . . . 11 𝑦 ∈ V
53, 4pm3.2i 266 . . . . . . . . . 10 (𝑥 ∈ V ∧ 𝑦 ∈ V)
6 opeqex 4067 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
75, 6mpbiri 166 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
87simpld 110 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
98adantr 270 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
109exlimivv 1824 . . . . . 6 (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
112, 10sylbi 119 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} → 𝐴 ∈ V)
12 df-eprel 4107 . . . . 5 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1311, 12eleq2s 2182 . . . 4 (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
141, 13sylbi 119 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
1514a1i 9 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
16 elex 2630 . . 3 (𝐴𝐵𝐴 ∈ V)
1716a1i 9 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
18 eleq12 2152 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
1918, 12brabga 4082 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
2019expcom 114 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
2115, 17, 20pm5.21ndd 656 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  cop 3444   class class class wbr 3837  {copab 3890   E cep 4105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-eprel 4107
This theorem is referenced by:  epelc  4109  efrirr  4171  smoiso  6049  ecidg  6336  ordiso2  6707  ltpiord  6857
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