Theorem opeqsn 4388

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 3898	. . 3 |- <. A , B >. = { { A } , { A , B } }
4	3	eqeq1i 2246	. 2 |- ( <. A , B >. = { C } <-> { { A } , { A , B } } = { C } )
5	1	snex 4317	. . 3 |- { A } e. _V
6		prexg 4344	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
7	1, 2, 6	mp2an 430	. . 3 |- { A , B } e. _V
8		opeqsn.3	. . 3 |- C e. _V
9	5, 7, 8	preqsn 3895	. 2 |- ( { { A } , { A , B } } = { C } <-> ( { A } = { A , B } /\ { A , B } = C ) )
10		eqcom 2240	. . . . 5 |- ( { A } = { A , B } <-> { A , B } = { A } )
11	1, 2, 1	preqsn 3895	. . . . 5 |- ( { A , B } = { A } <-> ( A = B /\ B = A ) )
12		eqcom 2240	. . . . . . 7 |- ( B = A <-> A = B )
13	12	anbi2i 461	. . . . . 6 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
14		anidm 400	. . . . . 6 |- ( ( A = B /\ A = B ) <-> A = B )
15	13, 14	bitri 184	. . . . 5 |- ( ( A = B /\ B = A ) <-> A = B )
16	10, 11, 15	3bitri 206	. . . 4 |- ( { A } = { A , B } <-> A = B )
17	16	anbi1i 462	. . 3 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ { A , B } = C ) )
18		dfsn2 3719	. . . . . . 7 |- { A } = { A , A }
19		preq2 3785	. . . . . . 7 |- ( A = B -> { A , A } = { A , B } )
20	18, 19	eqtr2id 2284	. . . . . 6 |- ( A = B -> { A , B } = { A } )
21	20	eqeq1d 2247	. . . . 5 |- ( A = B -> ( { A , B } = C <-> { A } = C ) )
22		eqcom 2240	. . . . 5 |- ( { A } = C <-> C = { A } )
23	21, 22	bitrdi 196	. . . 4 |- ( A = B -> ( { A , B } = C <-> C = { A } ) )
24	23	pm5.32i 458	. . 3 |- ( ( A = B /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
25	17, 24	bitri 184	. 2 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
26	4, 9, 25	3bitri 206	1 |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3705   {cpr 3706   <.cop 3708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714

This theorem is referenced by:  relop  4925