Theorem ideqg 4813

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 4791	. . . . 5 |- Rel _I
2	1	brrelex1i 4702	. . . 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 2256	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 299	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2771	. . . 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 2200	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2203	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4324	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4299	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 917	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4029   _I cid 4319

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666

This theorem is referenced by:  ideq  4814  ididg  4815  poleloe  5065