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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

Proof of Theorem epelg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3793 . . . 4
2 elopab 4023 . . . . . 6
3 vex 2577 . . . . . . . . . . 11
4 vex 2577 . . . . . . . . . . 11
53, 4pm3.2i 261 . . . . . . . . . 10
6 opeqex 4014 . . . . . . . . . 10
75, 6mpbiri 161 . . . . . . . . 9
87simpld 109 . . . . . . . 8
98adantr 265 . . . . . . 7
109exlimivv 1792 . . . . . 6
112, 10sylbi 118 . . . . 5
12 df-eprel 4054 . . . . 5
1311, 12eleq2s 2148 . . . 4
141, 13sylbi 118 . . 3
1514a1i 9 . 2
16 elex 2583 . . 3
1716a1i 9 . 2
18 eleq12 2118 . . . 4
1918, 12brabga 4029 . . 3
2019expcom 113 . 2
2115, 17, 20pm5.21ndd 631 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wb 102   wceq 1259  wex 1397   wcel 1409  cvv 2574  cop 3406   class class class wbr 3792  copab 3845   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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