Theorem ideqg 4771

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 4749	. . . . 5 |- Rel _I
2	1	brrelex1i 4663	. . . 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 2238	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 299	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2746	. . . 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 2182	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2185	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4287	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4263	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 916	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   _Vcvv 2735   class class class wbr 3998   _I cid 4282

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627

This theorem is referenced by:  ideq  4772  ididg  4773  poleloe  5020