Theorem epelc 4213

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

Step	Hyp	Ref	Expression
1		epelc.1	. 2 |- B e. _V
2		epelg 4212	. 2 |- ( B e. _V -> ( A _E B <-> A e. B ) )
3	1, 2	ax-mp 5	1 |- ( A _E B <-> A e. B )

Syntax hints:   <-> wb 104   e. wcel 1480   _Vcvv 2686   class class class wbr 3929   _E cep 4209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-eprel 4211

This theorem is referenced by:  epel  4214  epini  4910  ecid  6492  ordiso2  6920