Theorem opeqsn 4007

Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Hypotheses

Ref	Expression
opeqsn.1	|- A e. _V
opeqsn.2	|- B e. _V
opeqsn.3	|- C e. _V

Assertion

Ref	Expression
opeqsn	|- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) )

Proof of Theorem opeqsn

Step	Hyp	Ref	Expression
1		opeqsn.1	. . . 4 |- A e. _V
2		opeqsn.2	. . . 4 |- B e. _V
3	1, 2	dfop 3569	. . 3 |- <. A , B >. = { { A } , { A , B } }
4	3	eqeq1i 2088	. 2 |- ( <. A , B >. = { C } <-> { { A } , { A , B } } = { C } )
5	1	snex 3957	. . 3 |- { A } e. _V
6		prexg 3966	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
7	1, 2, 6	mp2an 416	. . 3 |- { A , B } e. _V
8		opeqsn.3	. . 3 |- C e. _V
9	5, 7, 8	preqsn 3567	. 2 |- ( { { A } , { A , B } } = { C } <-> ( { A } = { A , B } /\ { A , B } = C ) )
10		eqcom 2083	. . . . 5 |- ( { A } = { A , B } <-> { A , B } = { A } )
11	1, 2, 1	preqsn 3567	. . . . 5 |- ( { A , B } = { A } <-> ( A = B /\ B = A ) )
12		eqcom 2083	. . . . . . 7 |- ( B = A <-> A = B )
13	12	anbi2i 444	. . . . . 6 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
14		anidm 388	. . . . . 6 |- ( ( A = B /\ A = B ) <-> A = B )
15	13, 14	bitri 182	. . . . 5 |- ( ( A = B /\ B = A ) <-> A = B )
16	10, 11, 15	3bitri 204	. . . 4 |- ( { A } = { A , B } <-> A = B )
17	16	anbi1i 445	. . 3 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ { A , B } = C ) )
18		dfsn2 3412	. . . . . . 7 |- { A } = { A , A }
19		preq2 3470	. . . . . . 7 |- ( A = B -> { A , A } = { A , B } )
20	18, 19	syl5req 2126	. . . . . 6 |- ( A = B -> { A , B } = { A } )
21	20	eqeq1d 2089	. . . . 5 |- ( A = B -> ( { A , B } = C <-> { A } = C ) )
22		eqcom 2083	. . . . 5 |- ( { A } = C <-> C = { A } )
23	21, 22	syl6bb 194	. . . 4 |- ( A = B -> ( { A , B } = C <-> C = { A } ) )
24	23	pm5.32i 441	. . 3 |- ( ( A = B /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
25	17, 24	bitri 182	. 2 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
26	4, 9, 25	3bitri 204	1 |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   {csn 3398   {cpr 3399   <.cop 3401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407

This theorem is referenced by:  relop  4504