Theorem opeqsn 4077

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 3619	. . 3 |- <. A , B >. = { { A } , { A , B } }
4	3	eqeq1i 2095	. 2 |- ( <. A , B >. = { C } <-> { { A } , { A , B } } = { C } )
5	1	snex 4018	. . 3 |- { A } e. _V
6		prexg 4036	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
7	1, 2, 6	mp2an 417	. . 3 |- { A , B } e. _V
8		opeqsn.3	. . 3 |- C e. _V
9	5, 7, 8	preqsn 3617	. 2 |- ( { { A } , { A , B } } = { C } <-> ( { A } = { A , B } /\ { A , B } = C ) )
10		eqcom 2090	. . . . 5 |- ( { A } = { A , B } <-> { A , B } = { A } )
11	1, 2, 1	preqsn 3617	. . . . 5 |- ( { A , B } = { A } <-> ( A = B /\ B = A ) )
12		eqcom 2090	. . . . . . 7 |- ( B = A <-> A = B )
13	12	anbi2i 445	. . . . . 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 446	. . 3 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ { A , B } = C ) )
18		dfsn2 3458	. . . . . . 7 |- { A } = { A , A }
19		preq2 3518	. . . . . . 7 |- ( A = B -> { A , A } = { A , B } )
20	18, 19	syl5req 2133	. . . . . 6 |- ( A = B -> { A , B } = { A } )
21	20	eqeq1d 2096	. . . . 5 |- ( A = B -> ( { A , B } = C <-> { A } = C ) )
22		eqcom 2090	. . . . 5 |- ( { A } = C <-> C = { A } )
23	21, 22	syl6bb 194	. . . 4 |- ( A = B -> ( { A , B } = C <-> C = { A } ) )
24	23	pm5.32i 442	. . 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 1289   e. wcel 1438   _Vcvv 2619   {csn 3444   {cpr 3445   <.cop 3447

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453

This theorem is referenced by:  relop  4582