Theorem ideqg 4873

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 4851	. . . . 5 |- Rel _I
2	1	brrelex1i 4762	. . . 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 2292	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 299	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2811	. . . 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 2236	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2239	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4384	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4357	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 921	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   class class class wbr 4083   _I cid 4379

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726

This theorem is referenced by:  ideq  4874  ididg  4875  poleloe  5128