Theorem ideqg 4911

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 4889	. . . . 5 |- Rel _I
2	1	brrelex1i 4798	. . . 4 |- ( A _I B -> A e. _V )
3	2	adantl 277	. . 3 |- ( ( B e. V /\ A _I B ) -> A e. _V )
4		simpl 109	. . 3 |- ( ( B e. V /\ A _I B ) -> B e. V )
5	3, 4	jca 306	. 2 |- ( ( B e. V /\ A _I B ) -> ( A e. _V /\ B e. V ) )
6		eleq1 2297	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 299	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2827	. . . 4 |- ( A e. V -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ( B e. V /\ A = B ) -> A e. _V )
10		simpl 109	. . 3 |- ( ( B e. V /\ A = B ) -> B e. V )
11	9, 10	jca 306	. 2 |- ( ( B e. V /\ A = B ) -> ( A e. _V /\ B e. V ) )
12		eqeq1 2241	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2244	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4419	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4392	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 924	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   class class class wbr 4114   _I cid 4414

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761

This theorem is referenced by:  ideq  4912  ididg  4913  poleloe  5167