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Theorem preqsn 3697
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
StepHypRef Expression
1 dfsn2 3536 . . 3  |-  { C }  =  { C ,  C }
21eqeq2i 2148 . 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
63, 4, 5, 5preq12b 3692 . . 3  |-  ( { A ,  B }  =  { C ,  C } 
<->  ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C )
) )
7 oridm 746 . . . 4  |-  ( ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C ) )  <->  ( A  =  C  /\  B  =  C ) )
8 eqtr3 2157 . . . . . 6  |-  ( ( A  =  C  /\  B  =  C )  ->  A  =  B )
9 simpr 109 . . . . . 6  |-  ( ( A  =  C  /\  B  =  C )  ->  B  =  C )
108, 9jca 304 . . . . 5  |-  ( ( A  =  C  /\  B  =  C )  ->  ( A  =  B  /\  B  =  C ) )
11 eqtr 2155 . . . . . 6  |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )
12 simpr 109 . . . . . 6  |-  ( ( A  =  B  /\  B  =  C )  ->  B  =  C )
1311, 12jca 304 . . . . 5  |-  ( ( A  =  B  /\  B  =  C )  ->  ( A  =  C  /\  B  =  C ) )
1410, 13impbii 125 . . . 4  |-  ( ( A  =  C  /\  B  =  C )  <->  ( A  =  B  /\  B  =  C )
)
157, 14bitri 183 . . 3  |-  ( ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C ) )  <->  ( A  =  B  /\  B  =  C ) )
166, 15bitri 183 . 2  |-  ( { A ,  B }  =  { C ,  C } 
<->  ( A  =  B  /\  B  =  C ) )
172, 16bitri 183 1  |-  ( { A ,  B }  =  { C }  <->  ( A  =  B  /\  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   _Vcvv 2681   {csn 3522   {cpr 3523
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529
This theorem is referenced by:  opeqsn  4169  relop  4684
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