Theorem ideqg 4755

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 4733	. . . . 5 |- Rel _I
2	1	brrelex1i 4647	. . . 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 2229	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 297	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2737	. . . 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 2172	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2175	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4271	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4247	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 906	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   _I cid 4266

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611

This theorem is referenced by:  ideq  4756  ididg  4757  poleloe  5003