Theorem epse 4867

Description: The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
epse

Proof of Theorem epse

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		epel 4799	. . . . . . 7
2	1	bicomi 202	. . . . . 6
3	2	abbi2i 2590	. . . . 5
4		vex 3112	. . . . 5
5	3, 4	eqeltrri 2542	. . . 4
6		rabssab 3586	. . . 4
7	5, 6	ssexi 4597	. . 3
8	7	rgenw 2818	. 2
9		df-se 4844	. 2
10	8, 9	mpbir 209	1

Syntax hints:  e.wcel 1818  {cab 2442  A.wral 2807  {crab 2811  cvv 3109   class class class wbr 4452  cep 4794  Sewse 4841

This theorem is referenced by:  oieu  7985  oismo  7986  oiid  7987  cantnfp1lem3  8120  cantnfp1lem3OLD  8146  r0weon  8411  hsmexlem1  8827  omsinds  29299  tfrALTlem  29362  tfr1ALT  29363  tfr2ALT  29364  tfr3ALT  29365

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-ral 2812  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  df-se 4844