Theorem opeqsn 4229

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 3756	. . 3 |- <. A , B >. = { { A } , { A , B } }
4	3	eqeq1i 2173	. 2 |- ( <. A , B >. = { C } <-> { { A } , { A , B } } = { C } )
5	1	snex 4163	. . 3 |- { A } e. _V
6		prexg 4188	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
7	1, 2, 6	mp2an 423	. . 3 |- { A , B } e. _V
8		opeqsn.3	. . 3 |- C e. _V
9	5, 7, 8	preqsn 3754	. 2 |- ( { { A } , { A , B } } = { C } <-> ( { A } = { A , B } /\ { A , B } = C ) )
10		eqcom 2167	. . . . 5 |- ( { A } = { A , B } <-> { A , B } = { A } )
11	1, 2, 1	preqsn 3754	. . . . 5 |- ( { A , B } = { A } <-> ( A = B /\ B = A ) )
12		eqcom 2167	. . . . . . 7 |- ( B = A <-> A = B )
13	12	anbi2i 453	. . . . . 6 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
14		anidm 394	. . . . . 6 |- ( ( A = B /\ A = B ) <-> A = B )
15	13, 14	bitri 183	. . . . 5 |- ( ( A = B /\ B = A ) <-> A = B )
16	10, 11, 15	3bitri 205	. . . 4 |- ( { A } = { A , B } <-> A = B )
17	16	anbi1i 454	. . 3 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ { A , B } = C ) )
18		dfsn2 3589	. . . . . . 7 |- { A } = { A , A }
19		preq2 3653	. . . . . . 7 |- ( A = B -> { A , A } = { A , B } )
20	18, 19	eqtr2id 2211	. . . . . 6 |- ( A = B -> { A , B } = { A } )
21	20	eqeq1d 2174	. . . . 5 |- ( A = B -> ( { A , B } = C <-> { A } = C ) )
22		eqcom 2167	. . . . 5 |- ( { A } = C <-> C = { A } )
23	21, 22	bitrdi 195	. . . 4 |- ( A = B -> ( { A , B } = C <-> C = { A } ) )
24	23	pm5.32i 450	. . 3 |- ( ( A = B /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
25	17, 24	bitri 183	. 2 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
26	4, 9, 25	3bitri 205	1 |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2725   {csn 3575   {cpr 3576   <.cop 3578

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584

This theorem is referenced by:  relop  4753