Theorem epelg 4275

Description: The epsilon relation and membership are the same. General version of epel 4277. (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 3990	. . . 4 |- ( A _E B <-> <. A , B >. e. _E )
2		elopab 4243	. . . . . 6 |- ( <. A , B >. e. { <. x , y >. | x e. y } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ x e. y ) )
3		vex 2733	. . . . . . . . . . 11 |- x e. _V
4		vex 2733	. . . . . . . . . . 11 |- y e. _V
5	3, 4	pm3.2i 270	. . . . . . . . . 10 |- ( x e. _V /\ y e. _V )
6		opeqex 4234	. . . . . . . . . 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 1889	. . . . . 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 4274	. . . . 5 |- _E = { <. x , y >. | x e. y }
13	11, 12	eleq2s 2265	. . . 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 2741	. . 3 |- ( A e. B -> A e. _V )
17	16	a1i 9	. 2 |- ( B e. V -> ( A e. B -> A e. _V ) )
18		eleq12 2235	. . . 4 |- ( ( x = A /\ y = B ) -> ( x e. y <-> A e. B ) )
19	18, 12	brabga 4249	. . 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 700	1 |- ( B e. V -> ( A _E B <-> A e. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   <.cop 3586   class class class wbr 3989   {copab 4049   _E cep 4272

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-eprel 4274

This theorem is referenced by:  epelc  4276  efrirr  4338  smoiso  6281  ecidg  6577  ordiso2  7012  ltpiord  7281