Theorem preqsn 3776

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 3607	. . 3 |- { C } = { C , C }
2	1	eqeq2i 2188	. 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 3771	. . 3 |- ( { A , B } = { C , C } <-> ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) )
7		oridm 757	. . . 4 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = C /\ B = C ) )
8		eqtr3 2197	. . . . . 6 |- ( ( A = C /\ B = C ) -> A = B )
9		simpr 110	. . . . . 6 |- ( ( A = C /\ B = C ) -> B = C )
10	8, 9	jca 306	. . . . 5 |- ( ( A = C /\ B = C ) -> ( A = B /\ B = C ) )
11		eqtr 2195	. . . . . 6 |- ( ( A = B /\ B = C ) -> A = C )
12		simpr 110	. . . . . 6 |- ( ( A = B /\ B = C ) -> B = C )
13	11, 12	jca 306	. . . . 5 |- ( ( A = B /\ B = C ) -> ( A = C /\ B = C ) )
14	10, 13	impbii 126	. . . 4 |- ( ( A = C /\ B = C ) <-> ( A = B /\ B = C ) )
15	7, 14	bitri 184	. . 3 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = B /\ B = C ) )
16	6, 15	bitri 184	. 2 |- ( { A , B } = { C , C } <-> ( A = B /\ B = C ) )
17	2, 16	bitri 184	1 |- ( { A , B } = { C } <-> ( A = B /\ B = C ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   _Vcvv 2738   {csn 3593   {cpr 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600

This theorem is referenced by:  opeqsn  4253  relop  4778