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Theorem epelg 4018
Description: The epsilon relation and membership are the same. General version of epel 4020. (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 3756 . . . 4 (A E B ↔ ⟨A, B E )
2 elopab 3986 . . . . . 6 (⟨A, B {⟨x, y⟩ ∣ x y} ↔ xy(⟨A, B⟩ = ⟨x, y x y))
3 vex 2554 . . . . . . . . . . 11 x V
4 vex 2554 . . . . . . . . . . 11 y V
53, 4pm3.2i 257 . . . . . . . . . 10 (x V y V)
6 opeqex 3977 . . . . . . . . . 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 1773 . . . . . 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 4017 . . . . 5 E = {⟨x, y⟩ ∣ x y}
1311, 12eleq2s 2129 . . . 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 2560 . . 3 (A BA V)
1716a1i 9 . 2 (B 𝑉 → (A BA V))
18 eleq12 2099 . . . 4 ((x = A y = B) → (x yA B))
1918, 12brabga 3992 . . 3 ((A V B 𝑉) → (A E BA B))
2019expcom 109 . 2 (B 𝑉 → (A V → (A E BA B)))
2115, 17, 20pm5.21ndd 620 1 (B 𝑉 → (A E BA B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370   class class class wbr 3755  {copab 3808   E cep 4015
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-eprel 4017
This theorem is referenced by:  epelc  4019  smoiso  5858  ecidg  6106  ltpiord  6303
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