Theorem opeqsn 4967

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 4401	. . 3 |- <. A , B >. = { { A } , { A , B } }
4	3	eqeq1i 2627	. 2 |- ( <. A , B >. = { C } <-> { { A } , { A , B } } = { C } )
5		snex 4908	. . 3 |- { A } e. _V
6		prex 4909	. . 3 |- { A , B } e. _V
7		opeqsn.3	. . 3 |- C e. _V
8	5, 6, 7	preqsn 4393	. 2 |- ( { { A } , { A , B } } = { C } <-> ( { A } = { A , B } /\ { A , B } = C ) )
9		eqcom 2629	. . . . 5 |- ( { A } = { A , B } <-> { A , B } = { A } )
10	1, 2, 1	preqsn 4393	. . . . 5 |- ( { A , B } = { A } <-> ( A = B /\ B = A ) )
11		eqcom 2629	. . . . . . 7 |- ( B = A <-> A = B )
12	11	anbi2i 730	. . . . . 6 |- ( ( A = B /\ B = A ) <-> ( A = B /\ A = B ) )
13		anidm 676	. . . . . 6 |- ( ( A = B /\ A = B ) <-> A = B )
14	12, 13	bitri 264	. . . . 5 |- ( ( A = B /\ B = A ) <-> A = B )
15	9, 10, 14	3bitri 286	. . . 4 |- ( { A } = { A , B } <-> A = B )
16	15	anbi1i 731	. . 3 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ { A , B } = C ) )
17		dfsn2 4190	. . . . . . 7 |- { A } = { A , A }
18		preq2 4269	. . . . . . 7 |- ( A = B -> { A , A } = { A , B } )
19	17, 18	syl5req 2669	. . . . . 6 |- ( A = B -> { A , B } = { A } )
20	19	eqeq1d 2624	. . . . 5 |- ( A = B -> ( { A , B } = C <-> { A } = C ) )
21		eqcom 2629	. . . . 5 |- ( { A } = C <-> C = { A } )
22	20, 21	syl6bb 276	. . . 4 |- ( A = B -> ( { A , B } = C <-> C = { A } ) )
23	22	pm5.32i 669	. . 3 |- ( ( A = B /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
24	16, 23	bitri 264	. 2 |- ( ( { A } = { A , B } /\ { A , B } = C ) <-> ( A = B /\ C = { A } ) )
25	4, 8, 24	3bitri 286	1 |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   {cpr 4179   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  snopeqop  4969  propeqop  4970  relop  5272