Theorem ideqg 4762

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 4740	. . . . 5 |- Rel _I
2	1	brrelex1i 4654	. . . 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 2233	. . . . 5 |- ( A = B -> ( A e. V <-> B e. V ) )
7	6	biimparc 297	. . . 4 |- ( ( B e. V /\ A = B ) -> A e. V )
8		elex 2741	. . . 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 2177	. . 3 |- ( x = A -> ( x = y <-> A = y ) )
13		eqeq2 2180	. . 3 |- ( y = B -> ( A = y <-> A = B ) )
14		df-id 4278	. . 3 |- _I = { <. x , y >. | x = y }
15	12, 13, 14	brabg 4254	. 2 |- ( ( A e. _V /\ B e. V ) -> ( A _I B <-> A = B ) )
16	5, 11, 15	pm5.21nd 911	1 |- ( B e. V -> ( A _I B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   _I cid 4273

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618

This theorem is referenced by:  ideq  4763  ididg  4764  poleloe  5010