Theorem ideqg 4575

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 4553	. . . . 5 |- Rel _I
2	1	brrelexi 4468	. . . 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 2150	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 293	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2630	. . . 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 2094	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2097	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4111	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4087	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 863	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   _Vcvv 2619   class class class wbr 3837   _I cid 4106

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435

This theorem is referenced by:  ideq  4576  ididg  4577  poleloe  4818