Theorem ideqg 4698

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 4676	. . . . 5 |- Rel _I
2	1	brrelex1i 4590	. . . 4 |- ( A _I B -> A e. _V )
3	2	adantl 275	. . 3 |- ( ( B e. V /\ A _I B ) -> A e. _V )
4		simpl 108	. . 3 |- ( ( B e. V /\ A _I B ) -> B e. V )
5	3, 4	jca 304	. 2 |- ( ( B e. V /\ A _I B ) -> ( A e. _V /\ B e. V ) )
6		eleq1 2203	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 297	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2700	. . . 4 |- ( A e. V -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ( B e. V /\ A = B ) -> A e. _V )
10		simpl 108	. . 3 |- ( ( B e. V /\ A = B ) -> B e. V )
11	9, 10	jca 304	. 2 |- ( ( B e. V /\ A = B ) -> ( A e. _V /\ B e. V ) )
12		eqeq1 2147	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2150	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4223	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4199	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 902	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   _Vcvv 2689   class class class wbr 3937   _I cid 4218

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554

This theorem is referenced by:  ideq  4699  ididg  4700  poleloe  4946