Theorem epelg 4141

Description: The epsilon relation and membership are the same. General version of epel 4143. (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 3868	. . . 4 |- ( A _E B <-> <. A , B >. e. _E )
2		elopab 4109	. . . . . 6 |- ( <. A , B >. e. { <. x , y >. | x e. y } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ x e. y ) )
3		vex 2636	. . . . . . . . . . 11 |- x e. _V
4		vex 2636	. . . . . . . . . . 11 |- y e. _V
5	3, 4	pm3.2i 267	. . . . . . . . . 10 |- ( x e. _V /\ y e. _V )
6		opeqex 4100	. . . . . . . . . 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 271	. . . . . . 7 |- ( ( <. A , B >. = <. x , y >. /\ x e. y ) -> A e. _V )
10	9	exlimivv 1831	. . . . . 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 4140	. . . . 5 |- _E = { <. x , y >. | x e. y }
13	11, 12	eleq2s 2189	. . . 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 2644	. . 3 |- ( A e. B -> A e. _V )
17	16	a1i 9	. 2 |- ( B e. V -> ( A e. B -> A e. _V ) )
18		eleq12 2159	. . . 4 |- ( ( x = A /\ y = B ) -> ( x e. y <-> A e. B ) )
19	18, 12	brabga 4115	. . 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 659	1 |- ( B e. V -> ( A _E B <-> A e. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1296   E.wex 1433   e. wcel 1445   _Vcvv 2633   <.cop 3469   class class class wbr 3867   {copab 3920   _E cep 4138

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-eprel 4140

This theorem is referenced by:  epelc  4142  efrirr  4204  smoiso  6105  ecidg  6396  ordiso2  6808  ltpiord  6975