Theorem ideqg 4881

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 4859	. . . . 5 |- Rel _I
2	1	brrelex1i 4769	. . . 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 2294	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 299	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2814	. . . 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 2238	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2241	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4390	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4363	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 923	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   _I cid 4385

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732

This theorem is referenced by:  ideq  4882  ididg  4883  poleloe  5136