Theorem preqsn 3567

Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Hypotheses

Ref	Expression
preqsn.1	|- A e. _V
preqsn.2	|- B e. _V
preqsn.3	|- C e. _V

Assertion

Ref	Expression
preqsn	|- ( { A , B } = { C } <-> ( A = B /\ B = C ) )

Proof of Theorem preqsn

Step	Hyp	Ref	Expression
1		dfsn2 3412	. . 3 |- { C } = { C , C }
2	1	eqeq2i 2091	. 2 |- ( { A , B } = { C } <-> { A , B } = { C , C } )
3		preqsn.1	. . . 4 |- A e. _V
4		preqsn.2	. . . 4 |- B e. _V
5		preqsn.3	. . . 4 |- C e. _V
6	3, 4, 5, 5	preq12b 3562	. . 3 |- ( { A , B } = { C , C } <-> ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) )
7		oridm 706	. . . 4 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = C /\ B = C ) )
8		eqtr3 2100	. . . . . 6 |- ( ( A = C /\ B = C ) -> A = B )
9		simpr 108	. . . . . 6 |- ( ( A = C /\ B = C ) -> B = C )
10	8, 9	jca 300	. . . . 5 |- ( ( A = C /\ B = C ) -> ( A = B /\ B = C ) )
11		eqtr 2098	. . . . . 6 |- ( ( A = B /\ B = C ) -> A = C )
12		simpr 108	. . . . . 6 |- ( ( A = B /\ B = C ) -> B = C )
13	11, 12	jca 300	. . . . 5 |- ( ( A = B /\ B = C ) -> ( A = C /\ B = C ) )
14	10, 13	impbii 124	. . . 4 |- ( ( A = C /\ B = C ) <-> ( A = B /\ B = C ) )
15	7, 14	bitri 182	. . 3 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = B /\ B = C ) )
16	6, 15	bitri 182	. 2 |- ( { A , B } = { C , C } <-> ( A = B /\ B = C ) )
17	2, 16	bitri 182	1 |- ( { A , B } = { C } <-> ( A = B /\ B = C ) )

Syntax hints:   /\ wa 102   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   _Vcvv 2601   {csn 3398   {cpr 3399

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by:  opeqsn  4007  relop  4504