Theorem ideqg 4505

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ideqg	|- ( B e. V -> ( A _I B <-> A = B ) )

Proof of Theorem ideqg

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

Step	Hyp	Ref	Expression
1		reli 4483	. . . . 5 |- Rel _I
2	1	brrelexi 4402	. . . 4 |- ( A _I B -> A e. _V )
3	2	adantl 271	. . 3 |- ( ( B e. V /\ A _I B ) -> A e. _V )
4		simpl 107	. . 3 |- ( ( B e. V /\ A _I B ) -> B e. V )
5	3, 4	jca 300	. 2 |- ( ( B e. V /\ A _I B ) -> ( A e. _V /\ B e. V ) )
6		eleq1 2141	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 293	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2610	. . . 4 |- ( A e. V -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ( B e. V /\ A = B ) -> A e. _V )
10		simpl 107	. . 3 |- ( ( B e. V /\ A = B ) -> B e. V )
11	9, 10	jca 300	. 2 |- ( ( B e. V /\ A = B ) -> ( A e. _V /\ B e. V ) )
12		eqeq1 2087	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2090	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4048	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4024	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 858	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   _I cid 4043

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370

This theorem is referenced by:  ideq  4506  ididg  4507  poleloe  4744