Theorem epelc 4028

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 4027	. 2 |- ( B e. _V -> ( A _E B <-> A e. B ) )
3	1, 2	ax-mp 7	1 |- ( A _E B <-> A e. B )

Syntax hints:   <-> wb 98   e. wcel 1393   _Vcvv 2557   class class class wbr 3764   _E cep 4024

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026

This theorem is referenced by:  epel  4029  epini  4696  ecid  6169  ordiso2  6357