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Theorem epelg 4055
Description: The epsilon relation and membership are the same. General version of epel 4057. (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 3793 . . . 4 (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2 elopab 4023 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦))
3 vex 2577 . . . . . . . . . . 11 𝑥 ∈ V
4 vex 2577 . . . . . . . . . . 11 𝑦 ∈ V
53, 4pm3.2i 261 . . . . . . . . . 10 (𝑥 ∈ V ∧ 𝑦 ∈ V)
6 opeqex 4014 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
75, 6mpbiri 161 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
87simpld 109 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
98adantr 265 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
109exlimivv 1792 . . . . . 6 (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
112, 10sylbi 118 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} → 𝐴 ∈ V)
12 df-eprel 4054 . . . . 5 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1311, 12eleq2s 2148 . . . 4 (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
141, 13sylbi 118 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
1514a1i 9 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
16 elex 2583 . . 3 (𝐴𝐵𝐴 ∈ V)
1716a1i 9 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
18 eleq12 2118 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
1918, 12brabga 4029 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
2019expcom 113 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
2115, 17, 20pm5.21ndd 631 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  cop 3406   class class class wbr 3792  {copab 3845   E cep 4052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-eprel 4054
This theorem is referenced by:  epelc  4056  efrirr  4118  smoiso  5948  ecidg  6201  ordiso2  6415  ltpiord  6475
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