Theorem ideqg 4487

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 4465	. . . . 5 |- Rel _I
2	1	brrelexi 4384	. . . 4 |- ( A _I B -> A e. _V )
3	2	adantl 262	. . 3 |- ( ( B e. V /\ A _I B ) -> A e. _V )
4		simpl 102	. . 3 |- ( ( B e. V /\ A _I B ) -> B e. V )
5	3, 4	jca 290	. 2 |- ( ( B e. V /\ A _I B ) -> ( A e. _V /\ B e. V ) )
6		eleq1 2100	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 283	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2566	. . . 4 |- ( A e. V -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ( B e. V /\ A = B ) -> A e. _V )
10		simpl 102	. . 3 |- ( ( B e. V /\ A = B ) -> B e. V )
11	9, 10	jca 290	. 2 |- ( ( B e. V /\ A = B ) -> ( A e. _V /\ B e. V ) )
12		eqeq1 2046	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2049	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4030	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4006	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 825	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   class class class wbr 3764   _I cid 4025

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352

This theorem is referenced by:  ideq  4488  ididg  4489  poleloe  4724