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Theorem epelg 3997
Description: The epsilon relation and membership are the same. General version of epel 3999. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
epelg (B 𝑉 → (A E BA B))

Proof of Theorem epelg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3735 . . . 4 (A E B ↔ ⟨A, B E )
2 elopab 3965 . . . . . 6 (⟨A, B {⟨x, y⟩ ∣ x y} ↔ xy(⟨A, B⟩ = ⟨x, y x y))
3 vex 2534 . . . . . . . . . . 11 x V
4 vex 2534 . . . . . . . . . . 11 y V
53, 4pm3.2i 257 . . . . . . . . . 10 (x V y V)
6 opeqex 3956 . . . . . . . . . 10 (⟨A, B⟩ = ⟨x, y⟩ → ((A V B V) ↔ (x V y V)))
75, 6mpbiri 157 . . . . . . . . 9 (⟨A, B⟩ = ⟨x, y⟩ → (A V B V))
87simpld 105 . . . . . . . 8 (⟨A, B⟩ = ⟨x, y⟩ → A V)
98adantr 261 . . . . . . 7 ((⟨A, B⟩ = ⟨x, y x y) → A V)
109exlimivv 1754 . . . . . 6 (xy(⟨A, B⟩ = ⟨x, y x y) → A V)
112, 10sylbi 114 . . . . 5 (⟨A, B {⟨x, y⟩ ∣ x y} → A V)
12 df-eprel 3996 . . . . 5 E = {⟨x, y⟩ ∣ x y}
1311, 12eleq2s 2110 . . . 4 (⟨A, B E → A V)
141, 13sylbi 114 . . 3 (A E BA V)
1514a1i 9 . 2 (B 𝑉 → (A E BA V))
16 elex 2539 . . 3 (A BA V)
1716a1i 9 . 2 (B 𝑉 → (A BA V))
18 eleq12 2080 . . . 4 ((x = A y = B) → (x yA B))
1918, 12brabga 3971 . . 3 ((A V B 𝑉) → (A E BA B))
2019expcom 109 . 2 (B 𝑉 → (A V → (A E BA B)))
2115, 17, 20pm5.21ndd 608 1 (B 𝑉 → (A E BA B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226  wex 1358   wcel 1370  Vcvv 2531  cop 3349   class class class wbr 3734  {copab 3787   E cep 3994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-eprel 3996
This theorem is referenced by:  epelc  3998  smoiso  5835  ecidg  6077  ltpiord  6173
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