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Theorem epse 4557
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 4489 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 194 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2546 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 2951 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2506 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 rabssab 3422 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
75, 6ssexi 4340 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
87rgenw 2765 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
9 df-se 4534 . 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 1725   {cab 2421   A.wral 2697   {crab 2701   _Vcvv 2948   class class class wbr 4204    _E cep 4484   Se wse 4531
This theorem is referenced by:  oieu  7500  oismo  7501  oiid  7502  cantnfp1lem3  7628  r0weon  7886  hsmexlem1  8298  omsinds  25486  tfrALTlem  25549  tfr1ALT  25550  tfr2ALT  25551  tfr3ALT  25552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-eprel 4486  df-se 4534
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