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Theorem epse 4499
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 4431 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 194 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2491 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 2895 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2451 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 rabssab 3366 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
75, 6ssexi 4282 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
87rgenw 2709 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
9 df-se 4476 . 2  |-  (  _E Se 
A  <->  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V )
108, 9mpbir 201 1  |-  _E Se  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   {cab 2366   A.wral 2642   {crab 2646   _Vcvv 2892   class class class wbr 4146    _E cep 4426   Se wse 4473
This theorem is referenced by:  oieu  7434  oismo  7435  oiid  7436  cantnfp1lem3  7562  r0weon  7820  hsmexlem1  8232  omsinds  25236  tfr1ALT  25294  tfr2ALT  25295  tfr3ALT  25296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-eprel 4428  df-se 4476
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