Theorem epelg 4268

Description: The epsilon relation and membership are the same. General version of epel 4270. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
epelg	|- ( B e. V -> ( A _E B <-> A e. B ) )

Proof of Theorem epelg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 3983	. . . 4 |- ( A _E B <-> <. A , B >. e. _E )
2		elopab 4236	. . . . . 6 |- ( <. A , B >. e. { <. x , y >. | x e. y } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ x e. y ) )
3		vex 2729	. . . . . . . . . . 11 |- x e. _V
4		vex 2729	. . . . . . . . . . 11 |- y e. _V
5	3, 4	pm3.2i 270	. . . . . . . . . 10 |- ( x e. _V /\ y e. _V )
6		opeqex 4227	. . . . . . . . . 10 |- ( <. A , B >. = <. x , y >. -> ( ( A e. _V /\ B e. _V ) <-> ( x e. _V /\ y e. _V ) ) )
7	5, 6	mpbiri 167	. . . . . . . . 9 |- ( <. A , B >. = <. x , y >. -> ( A e. _V /\ B e. _V ) )
8	7	simpld 111	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> A e. _V )
9	8	adantr 274	. . . . . . 7 |- ( ( <. A , B >. = <. x , y >. /\ x e. y ) -> A e. _V )
10	9	exlimivv 1884	. . . . . 6 |- ( E. x E. y ( <. A , B >. = <. x , y >. /\ x e. y ) -> A e. _V )
11	2, 10	sylbi 120	. . . . 5 |- ( <. A , B >. e. { <. x , y >. | x e. y } -> A e. _V )
12		df-eprel 4267	. . . . 5 |- _E = { <. x , y >. | x e. y }
13	11, 12	eleq2s 2261	. . . 4 |- ( <. A , B >. e. _E -> A e. _V )
14	1, 13	sylbi 120	. . 3 |- ( A _E B -> A e. _V )
15	14	a1i 9	. 2 |- ( B e. V -> ( A _E B -> A e. _V ) )
16		elex 2737	. . 3 |- ( A e. B -> A e. _V )
17	16	a1i 9	. 2 |- ( B e. V -> ( A e. B -> A e. _V ) )
18		eleq12 2231	. . . 4 |- ( ( x = A /\ y = B ) -> ( x e. y <-> A e. B ) )
19	18, 12	brabga 4242	. . 3 |- ( ( A e. _V /\ B e. V ) -> ( A _E B <-> A e. B ) )
20	19	expcom 115	. 2 |- ( B e. V -> ( A e. _V -> ( A _E B <-> A e. B ) ) )
21	15, 17, 20	pm5.21ndd 695	1 |- ( B e. V -> ( A _E B <-> A e. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   <.cop 3579   class class class wbr 3982   {copab 4042   _E cep 4265

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-eprel 4267

This theorem is referenced by:  epelc  4269  efrirr  4331  smoiso  6270  ecidg  6565  ordiso2  7000  ltpiord  7260