Theorem opeqsn 3989

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 3548	. . 3 |- <. A , B >. = { { A } , { A , B } }
4	3	eqeq1i 2047	. 2 |- ( <. A , B >. = { C } <-> { { A } , { A , B } } = { C } )
5		snexgOLD 3935	. . . 4 |- ( A e. _V -> { A } e. _V )
6	1, 5	ax-mp 7	. . 3 |- { A } e. _V
7		prexgOLD 3946	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
8	1, 2, 7	mp2an 402	. . 3 |- { A , B } e. _V
9		opeqsn.3	. . 3 |- C e. _V
10	6, 8, 9	preqsn 3546	. 2 |- ( { { A } , { A , B } } = { C } <-> ( { A } = { A , B } /\ { A , B } = C ) )
11		eqcom 2042	. . . . 5 |- ( { A } = { A , B } <-> { A , B } = { A } )
12	1, 2, 1	preqsn 3546	. . . . 5 |- ( { A , B } = { A } <-> ( A = B /\ B = A ) )
13		eqcom 2042	. . . . . . 7 |- ( B = A <-> A = B )
14	13	anbi2i 430	. . . . . 6 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
15		anidm 376	. . . . . 6 |- ( ( A = B /\ A = B ) <-> A = B )
16	14, 15	bitri 173	. . . . 5 |- ( ( A = B /\ B = A ) <-> A = B )
17	11, 12, 16	3bitri 195	. . . 4 |- ( { A } = { A , B } <-> A = B )
18	17	anbi1i 431	. . 3 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ { A , B } = C ) )
19		dfsn2 3389	. . . . . . 7 |- { A } = { A , A }
20		preq2 3448	. . . . . . 7 |- ( A = B -> { A , A } = { A , B } )
21	19, 20	syl5req 2085	. . . . . 6 |- ( A = B -> { A , B } = { A } )
22	21	eqeq1d 2048	. . . . 5 |- ( A = B -> ( { A , B } = C <-> { A } = C ) )
23		eqcom 2042	. . . . 5 |- ( { A } = C <-> C = { A } )
24	22, 23	syl6bb 185	. . . 4 |- ( A = B -> ( { A , B } = C <-> C = { A } ) )
25	24	pm5.32i 427	. . 3 |- ( ( A = B /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
26	18, 25	bitri 173	. 2 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
27	4, 10, 26	3bitri 195	1 |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   {csn 3375   {cpr 3376   <.cop 3378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384

This theorem is referenced by:  relop  4486