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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.
StepHypRef Expression
1 epel 4799 . . . . . . 7
21bicomi 202 . . . . . 6
32abbi2i 2590 . . . . 5
4 vex 3112 . . . . 5
53, 4eqeltrri 2542 . . . 4
6 rabssab 3586 . . . 4
75, 6ssexi 4597 . . 3
87rgenw 2818 . 2
9 df-se 4844 . 2
108, 9mpbir 209 1
 Colors of variables: wff setvar class 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
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