Theorem preqsn 3710

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 3546	. . 3 |- { C } = { C , C }
2	1	eqeq2i 2151	. 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 3705	. . 3 |- ( { A , B } = { C , C } <-> ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) )
7		oridm 747	. . . 4 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = C /\ B = C ) )
8		eqtr3 2160	. . . . . 6 |- ( ( A = C /\ B = C ) -> A = B )
9		simpr 109	. . . . . 6 |- ( ( A = C /\ B = C ) -> B = C )
10	8, 9	jca 304	. . . . 5 |- ( ( A = C /\ B = C ) -> ( A = B /\ B = C ) )
11		eqtr 2158	. . . . . 6 |- ( ( A = B /\ B = C ) -> A = C )
12		simpr 109	. . . . . 6 |- ( ( A = B /\ B = C ) -> B = C )
13	11, 12	jca 304	. . . . 5 |- ( ( A = B /\ B = C ) -> ( A = C /\ B = C ) )
14	10, 13	impbii 125	. . . 4 |- ( ( A = C /\ B = C ) <-> ( A = B /\ B = C ) )
15	7, 14	bitri 183	. . 3 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = B /\ B = C ) )
16	6, 15	bitri 183	. 2 |- ( { A , B } = { C , C } <-> ( A = B /\ B = C ) )
17	2, 16	bitri 183	1 |- ( { A , B } = { C } <-> ( A = B /\ B = C ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1332   e. wcel 1481   _Vcvv 2689   {csn 3532   {cpr 3533

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539

This theorem is referenced by:  opeqsn  4182  relop  4697