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Theorem epelg 4172
Description: The epsilon relation and membership are the same. General version of epel 4174. (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 3896 . . . 4 (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2 elopab 4140 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦))
3 vex 2660 . . . . . . . . . . 11 𝑥 ∈ V
4 vex 2660 . . . . . . . . . . 11 𝑦 ∈ V
53, 4pm3.2i 268 . . . . . . . . . 10 (𝑥 ∈ V ∧ 𝑦 ∈ V)
6 opeqex 4131 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
75, 6mpbiri 167 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
87simpld 111 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
98adantr 272 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
109exlimivv 1850 . . . . . 6 (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
112, 10sylbi 120 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} → 𝐴 ∈ V)
12 df-eprel 4171 . . . . 5 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1311, 12eleq2s 2209 . . . 4 (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
141, 13sylbi 120 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
1514a1i 9 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
16 elex 2668 . . 3 (𝐴𝐵𝐴 ∈ V)
1716a1i 9 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
18 eleq12 2179 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
1918, 12brabga 4146 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
2019expcom 115 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
2115, 17, 20pm5.21ndd 677 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2657  cop 3496   class class class wbr 3895  {copab 3948   E cep 4169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-eprel 4171
This theorem is referenced by:  epelc  4173  efrirr  4235  smoiso  6153  ecidg  6447  ordiso2  6872  ltpiord  7075
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