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Theorem epse 4160
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 4110 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 130 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2202 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 2622 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2161 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 rabssab 3106 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
75, 6ssexi 3969 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
87rgenw 2430 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
9 df-se 4151 . 2  |-  (  _E Se 
A  <->  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V )
108, 9mpbir 144 1  |-  _E Se  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1438   {cab 2074   A.wral 2359   {crab 2363   _Vcvv 2619   class class class wbr 3837    _E cep 4105   Se wse 4147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-eprel 4107  df-se 4151
This theorem is referenced by: (None)
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