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.

Step	Hyp	Ref	Expression
1		df-br 4453	. . . 4
2		elopab 4760	. . . . . 6
3		vex 3112	. . . . . . . . . . 11
4		vex 3112	. . . . . . . . . . 11
5	3, 4	pm3.2i 455	. . . . . . . . . 10
6		opeqex 4743	. . . . . . . . . 10
7	5, 6	mpbiri 233	. . . . . . . . 9
8	7	simpld 459	. . . . . . . 8
9	8	adantr 465	. . . . . . 7
10	9	exlimivv 1723	. . . . . 6
11	2, 10	sylbi 195	. . . . 5
12		df-eprel 4796	. . . . 5
13	11, 12	eleq2s 2565	. . . 4
14	1, 13	sylbi 195	. . 3
15	14	a1i 11	. 2
16		elex 3118	. . 3
17	16	a1i 11	. 2
18		eleq12 2533	. . . 4
19	18, 12	brabga 4766	. . 3
20	19	expcom 435	. 2
21	15, 17, 20	pm5.21ndd 354	1

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