Theorem preqsn 3537

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 3381	. . 3 |- { C } = { C , C }
2	1	eqeq2i 2047	. 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 3532	. . 3 |- ( {A , B } = { C , C } <-> ((A = C /\ B = C) \/ (A = C /\ B = C)))
7		oridm 673	. . . 4 |- (((A = C /\ B = C) \/ (A = C /\ B = C)) <-> (A = C /\ B = C))
8		eqtr3 2056	. . . . . 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 2054	. . . . . 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 628   = wceq 1242   e. wcel 1390   _Vcvv 2551   {csn 3367   {cpr 3368

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374

This theorem is referenced by:  opeqsn  3980  relop  4429