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Theorem epse 4702
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 4634 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 202 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2553 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 2974 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2513 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 rabssab 3438 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
75, 6ssexi 4436 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
87rgenw 2782 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
9 df-se 4679 . 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 1756   {cab 2428   A.wral 2714   {crab 2718   _Vcvv 2971   class class class wbr 4291    _E cep 4629   Se wse 4676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-eprel 4631  df-se 4679
This theorem is referenced by:  oieu  7752  oismo  7753  oiid  7754  cantnfp1lem3  7887  cantnfp1lem3OLD  7913  r0weon  8178  hsmexlem1  8594  omsinds  27679  tfrALTlem  27742  tfr1ALT  27743  tfr2ALT  27744  tfr3ALT  27745
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