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Theorem epelg 4497
 Description: The epsilon relation and membership are the same. General version of epel 4499. (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 4215 . . . 4
2 elopab 4464 . . . . . 6
3 vex 2961 . . . . . . . . . . 11
4 vex 2961 . . . . . . . . . . 11
53, 4pm3.2i 443 . . . . . . . . . 10
6 opeqex 4449 . . . . . . . . . 10
75, 6mpbiri 226 . . . . . . . . 9
87simpld 447 . . . . . . . 8
98adantr 453 . . . . . . 7
109exlimivv 1646 . . . . . 6
112, 10sylbi 189 . . . . 5
12 df-eprel 4496 . . . . 5
1311, 12eleq2s 2530 . . . 4
141, 13sylbi 189 . . 3
1514a1i 11 . 2
16 elex 2966 . . 3
1716a1i 11 . 2
18 eleq12 2500 . . . 4
1918, 12brabga 4471 . . 3
2019expcom 426 . 2
2115, 17, 20pm5.21ndd 345 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726  cvv 2958  cop 3819   class class class wbr 4214  copab 4267   cep 4494 This theorem is referenced by:  epelc  4498  efrirr  4565  efrn2lp  4566  epne3  4763  cnfcomlem  7658  fpwwe2lem6  8512  ltpiord  8766  orvcelval  24728  predep  25469 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-eprel 4496
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