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Theorem epelg 4797
 Description: The epsilon relation and membership are the same. General version of epel 4799. (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 4453 . . . 4
2 elopab 4760 . . . . . 6
3 vex 3112 . . . . . . . . . . 11
4 vex 3112 . . . . . . . . . . 11
53, 4pm3.2i 455 . . . . . . . . . 10
6 opeqex 4743 . . . . . . . . . 10
75, 6mpbiri 233 . . . . . . . . 9
87simpld 459 . . . . . . . 8
98adantr 465 . . . . . . 7
109exlimivv 1723 . . . . . 6
112, 10sylbi 195 . . . . 5
12 df-eprel 4796 . . . . 5
1311, 12eleq2s 2565 . . . 4
141, 13sylbi 195 . . 3
1514a1i 11 . 2
16 elex 3118 . . 3
1716a1i 11 . 2
18 eleq12 2533 . . . 4
1918, 12brabga 4766 . . 3
2019expcom 435 . 2
2115, 17, 20pm5.21ndd 354 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  <.cop 4035   class class class wbr 4452  {copab 4509   cep 4794 This theorem is referenced by:  epelc  4798  efrirr  4865  efrn2lp  4866  epne3  6616  cnfcomlem  8164  cnfcomlemOLD  8172  fpwwe2lem6  9034  ltpiord  9286  orvcelval  28407  predep  29272 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-eprel 4796
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