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Theorem epse 4392
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 4324 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 193 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2407 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 2804 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2367 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 dfrab2 3456 . . . . 5  |-  { y  e.  A  |  y  _E  x }  =  ( { y  |  y  _E  x }  i^i  A )
7 inss1 3402 . . . . 5  |-  ( { y  |  y  _E  x }  i^i  A
)  C_  { y  |  y  _E  x }
86, 7eqsstri 3221 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
95, 8ssexi 4175 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
109rgenw 2623 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
11 df-se 4369 . 2  |-  (  _E Se 
A  <->  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V )
1210, 11mpbir 200 1  |-  _E Se  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   {cab 2282   A.wral 2556   {crab 2560   _Vcvv 2801    i^i cin 3164   class class class wbr 4039    _E cep 4319   Se wse 4366
This theorem is referenced by:  oieu  7270  oismo  7271  oiid  7272  cantnfp1lem3  7398  r0weon  7656  hsmexlem1  8068  omsinds  24290  tfr1ALT  24348  tfr2ALT  24349  tfr3ALT  24350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-eprel 4321  df-se 4369
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