Theorem preqsn 3546

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 3389	. . 3 |- { C } = { C , C }
2	1	eqeq2i 2050	. 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 3541	. . 3 |- ( { A , B } = { C , C } <-> ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) )
7		oridm 674	. . . 4 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = C /\ B = C ) )
8		eqtr3 2059	. . . . . 6 |- ( ( A = C /\ B = C ) -> A = B )
9		simpr 103	. . . . . 6 |- ( ( A = C /\ B = C ) -> B = C )
10	8, 9	jca 290	. . . . 5 |- ( ( A = C /\ B = C ) -> ( A = B /\ B = C ) )
11		eqtr 2057	. . . . . 6 |- ( ( A = B /\ B = C ) -> A = C )
12		simpr 103	. . . . . 6 |- ( ( A = B /\ B = C ) -> B = C )
13	11, 12	jca 290	. . . . 5 |- ( ( A = B /\ B = C ) -> ( A = C /\ B = C ) )
14	10, 13	impbii 117	. . . 4 |- ( ( A = C /\ B = C ) <-> ( A = B /\ B = C ) )
15	7, 14	bitri 173	. . 3 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = B /\ B = C ) )
16	6, 15	bitri 173	. 2 |- ( { A , B } = { C , C } <-> ( A = B /\ B = C ) )
17	2, 16	bitri 173	1 |- ( { A , B } = { C } <-> ( A = B /\ B = C ) )

Syntax hints:   /\ wa 97   <-> wb 98   \/ wo 629   = wceq 1243   e. wcel 1393   _Vcvv 2557   {csn 3375   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  opeqsn  3989  relop  4486