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Theorem opeqsn 4142
Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
opeqsn.1  |-  A  e. 
_V
opeqsn.2  |-  B  e. 
_V
opeqsn.3  |-  C  e. 
_V
Assertion
Ref Expression
opeqsn  |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )

Proof of Theorem opeqsn
StepHypRef Expression
1 opeqsn.1 . . . 4  |-  A  e. 
_V
2 opeqsn.2 . . . 4  |-  B  e. 
_V
31, 2dfop 3672 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43eqeq1i 2123 . 2  |-  ( <. A ,  B >.  =  { C }  <->  { { A } ,  { A ,  B } }  =  { C } )
51snex 4077 . . 3  |-  { A }  e.  _V
6 prexg 4101 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
71, 2, 6mp2an 420 . . 3  |-  { A ,  B }  e.  _V
8 opeqsn.3 . . 3  |-  C  e. 
_V
95, 7, 8preqsn 3670 . 2  |-  ( { { A } ,  { A ,  B } }  =  { C } 
<->  ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C
) )
10 eqcom 2117 . . . . 5  |-  ( { A }  =  { A ,  B }  <->  { A ,  B }  =  { A } )
111, 2, 1preqsn 3670 . . . . 5  |-  ( { A ,  B }  =  { A }  <->  ( A  =  B  /\  B  =  A ) )
12 eqcom 2117 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
1312anbi2i 450 . . . . . 6  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
14 anidm 391 . . . . . 6  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
1513, 14bitri 183 . . . . 5  |-  ( ( A  =  B  /\  B  =  A )  <->  A  =  B )
1610, 11, 153bitri 205 . . . 4  |-  ( { A }  =  { A ,  B }  <->  A  =  B )
1716anbi1i 451 . . 3  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  { A ,  B }  =  C ) )
18 dfsn2 3509 . . . . . . 7  |-  { A }  =  { A ,  A }
19 preq2 3569 . . . . . . 7  |-  ( A  =  B  ->  { A ,  A }  =  { A ,  B }
)
2018, 19syl5req 2161 . . . . . 6  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2120eqeq1d 2124 . . . . 5  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  { A }  =  C ) )
22 eqcom 2117 . . . . 5  |-  ( { A }  =  C  <-> 
C  =  { A } )
2321, 22syl6bb 195 . . . 4  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  C  =  { A } ) )
2423pm5.32i 447 . . 3  |-  ( ( A  =  B  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
2517, 24bitri 183 . 2  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
264, 9, 253bitri 205 1  |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   _Vcvv 2658   {csn 3495   {cpr 3496   <.cop 3498
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504
This theorem is referenced by:  relop  4657
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