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Theorem epse 4105
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 4055 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 130 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2194 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 2605 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2153 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 rabssab 3082 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
75, 6ssexi 3924 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
87rgenw 2419 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
9 df-se 4096 . 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 1434   {cab 2068   A.wral 2349   {crab 2353   _Vcvv 2602   class class class wbr 3793    _E cep 4050   Se wse 4092
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-eprel 4052  df-se 4096
This theorem is referenced by: (None)
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