Theorem epelg 4321

Description: The epsilon relation and membership are the same. General version of epel 4323. (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 4030	. . . 4 |- ( A _E B <-> <. A , B >. e. _E )
2		elopab 4288	. . . . . 6 |- ( <. A , B >. e. { <. x , y >. | x e. y } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ x e. y ) )
3		vex 2763	. . . . . . . . . . 11 |- x e. _V
4		vex 2763	. . . . . . . . . . 11 |- y e. _V
5	3, 4	pm3.2i 272	. . . . . . . . . 10 |- ( x e. _V /\ y e. _V )
6		opeqex 4278	. . . . . . . . . 10 |- ( <. A , B >. = <. x , y >. -> ( ( A e. _V /\ B e. _V ) <-> ( x e. _V /\ y e. _V ) ) )
7	5, 6	mpbiri 168	. . . . . . . . 9 |- ( <. A , B >. = <. x , y >. -> ( A e. _V /\ B e. _V ) )
8	7	simpld 112	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> A e. _V )
9	8	adantr 276	. . . . . . 7 |- ( ( <. A , B >. = <. x , y >. /\ x e. y ) -> A e. _V )
10	9	exlimivv 1908	. . . . . 6 |- ( E. x E. y ( <. A , B >. = <. x , y >. /\ x e. y ) -> A e. _V )
11	2, 10	sylbi 121	. . . . 5 |- ( <. A , B >. e. { <. x , y >. | x e. y } -> A e. _V )
12		df-eprel 4320	. . . . 5 |- _E = { <. x , y >. | x e. y }
13	11, 12	eleq2s 2288	. . . 4 |- ( <. A , B >. e. _E -> A e. _V )
14	1, 13	sylbi 121	. . 3 |- ( A _E B -> A e. _V )
15	14	a1i 9	. 2 |- ( B e. V -> ( A _E B -> A e. _V ) )
16		elex 2771	. . 3 |- ( A e. B -> A e. _V )
17	16	a1i 9	. 2 |- ( B e. V -> ( A e. B -> A e. _V ) )
18		eleq12 2258	. . . 4 |- ( ( x = A /\ y = B ) -> ( x e. y <-> A e. B ) )
19	18, 12	brabga 4294	. . 3 |- ( ( A e. _V /\ B e. V ) -> ( A _E B <-> A e. B ) )
20	19	expcom 116	. 2 |- ( B e. V -> ( A e. _V -> ( A _E B <-> A e. B ) ) )
21	15, 17, 20	pm5.21ndd 706	1 |- ( B e. V -> ( A _E B <-> A e. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   <.cop 3621   class class class wbr 4029   {copab 4089   _E cep 4318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-eprel 4320

This theorem is referenced by:  epelc  4322  efrirr  4384  smoiso  6355  ecidg  6653  ordiso2  7094  ltpiord  7379