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Theorem epse 4862
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 4794 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 202 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2600 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 3116 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2552 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 rabssab 3587 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
75, 6ssexi 4592 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
87rgenw 2825 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
9 df-se 4839 . 2  |-  (  _E Se 
A  <->  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V )
108, 9mpbir 209 1  |-  _E Se  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   {cab 2452   A.wral 2814   {crab 2818   _Vcvv 3113   class class class wbr 4447    _E cep 4789   Se wse 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-eprel 4791  df-se 4839
This theorem is referenced by:  oieu  7964  oismo  7965  oiid  7966  cantnfp1lem3  8099  cantnfp1lem3OLD  8125  r0weon  8390  hsmexlem1  8806  omsinds  28904  tfrALTlem  28967  tfr1ALT  28968  tfr2ALT  28969  tfr3ALT  28970
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