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Theorem epse 4837
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  |-  _E Se  A

Proof of Theorem epse
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epel 4768 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 205 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2562 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 3090 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2514 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 rabssab 3554 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
75, 6ssexi 4570 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
87rgenw 2793 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
9 df-se 4814 . 2  |-  (  _E Se 
A  <->  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V )
108, 9mpbir 212 1  |-  _E Se  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1870   {cab 2414   A.wral 2782   {crab 2786   _Vcvv 3087   class class class wbr 4426    _E cep 4763   Se wse 4811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-eprel 4765  df-se 4814
This theorem is referenced by:  omsinds  6725  tfr1ALT  7126  tfr2ALT  7127  tfr3ALT  7128  oieu  8054  oismo  8055  oiid  8056  cantnfp1lem3  8184  r0weon  8442  hsmexlem1  8854
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