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Theorem epelc 4118
Description: The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
epelc.1  |-  B  e. 
_V
Assertion
Ref Expression
epelc  |-  ( A  _E  B  <->  A  e.  B )

Proof of Theorem epelc
StepHypRef Expression
1 epelc.1 . 2  |-  B  e. 
_V
2 epelg 4117 . 2  |-  ( B  e.  _V  ->  ( A  _E  B  <->  A  e.  B ) )
31, 2ax-mp 7 1  |-  ( A  _E  B  <->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1438   _Vcvv 2619   class class class wbr 3845    _E cep 4114
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 3957  ax-pow 4009  ax-pr 4036
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-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-eprel 4116
This theorem is referenced by:  epel  4119  epini  4803  ecid  6355  ordiso2  6728
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