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Theorem preqsn 4166
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 4001 . . 3  |-  { C }  =  { C ,  C }
21eqeq2i 2472 . 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 4159 . . 3  |-  ( { A ,  B }  =  { C ,  C } 
<->  ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C )
) )
7 oridm 514 . . . 4  |-  ( ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C ) )  <->  ( A  =  C  /\  B  =  C ) )
8 eqtr3 2482 . . . . . 6  |-  ( ( A  =  C  /\  B  =  C )  ->  A  =  B )
9 simpr 461 . . . . . 6  |-  ( ( A  =  C  /\  B  =  C )  ->  B  =  C )
108, 9jca 532 . . . . 5  |-  ( ( A  =  C  /\  B  =  C )  ->  ( A  =  B  /\  B  =  C ) )
11 eqtr 2480 . . . . . 6  |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )
12 simpr 461 . . . . . 6  |-  ( ( A  =  B  /\  B  =  C )  ->  B  =  C )
1311, 12jca 532 . . . . 5  |-  ( ( A  =  B  /\  B  =  C )  ->  ( A  =  C  /\  B  =  C ) )
1410, 13impbii 188 . . . 4  |-  ( ( A  =  C  /\  B  =  C )  <->  ( A  =  B  /\  B  =  C )
)
157, 14bitri 249 . . 3  |-  ( ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C ) )  <->  ( A  =  B  /\  B  =  C ) )
166, 15bitri 249 . 2  |-  ( { A ,  B }  =  { C ,  C } 
<->  ( A  =  B  /\  B  =  C ) )
172, 16bitri 249 1  |-  ( { A ,  B }  =  { C }  <->  ( A  =  B  /\  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   {csn 3988   {cpr 3990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3444  df-sn 3989  df-pr 3991
This theorem is referenced by:  opeqsn  4698  relop  5101  hash2prde  12300  symg2bas  16025
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