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Theorem preqsn 3981
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 3829 . . 3  |-  { C }  =  { C ,  C }
21eqeq2i 2447 . 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 3975 . . 3  |-  ( { A ,  B }  =  { C ,  C } 
<->  ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C )
) )
7 oridm 502 . . . 4  |-  ( ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C ) )  <->  ( A  =  C  /\  B  =  C ) )
8 eqtr3 2456 . . . . . 6  |-  ( ( A  =  C  /\  B  =  C )  ->  A  =  B )
9 simpr 449 . . . . . 6  |-  ( ( A  =  C  /\  B  =  C )  ->  B  =  C )
108, 9jca 520 . . . . 5  |-  ( ( A  =  C  /\  B  =  C )  ->  ( A  =  B  /\  B  =  C ) )
11 eqtr 2454 . . . . . 6  |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )
12 simpr 449 . . . . . 6  |-  ( ( A  =  B  /\  B  =  C )  ->  B  =  C )
1311, 12jca 520 . . . . 5  |-  ( ( A  =  B  /\  B  =  C )  ->  ( A  =  C  /\  B  =  C ) )
1410, 13impbii 182 . . . 4  |-  ( ( A  =  C  /\  B  =  C )  <->  ( A  =  B  /\  B  =  C )
)
157, 14bitri 242 . . 3  |-  ( ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C ) )  <->  ( A  =  B  /\  B  =  C ) )
166, 15bitri 242 . 2  |-  ( { A ,  B }  =  { C ,  C } 
<->  ( A  =  B  /\  B  =  C ) )
172, 16bitri 242 1  |-  ( { A ,  B }  =  { C }  <->  ( A  =  B  /\  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2957   {csn 3815   {cpr 3816
This theorem is referenced by:  opeqsn  4453  relop  5024  hash2prde  11689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-un 3326  df-sn 3821  df-pr 3822
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