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Theorem opeqsn 4077
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 3619 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43eqeq1i 2095 . 2  |-  ( <. A ,  B >.  =  { C }  <->  { { A } ,  { A ,  B } }  =  { C } )
51snex 4018 . . 3  |-  { A }  e.  _V
6 prexg 4036 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
71, 2, 6mp2an 417 . . 3  |-  { A ,  B }  e.  _V
8 opeqsn.3 . . 3  |-  C  e. 
_V
95, 7, 8preqsn 3617 . 2  |-  ( { { A } ,  { A ,  B } }  =  { C } 
<->  ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C
) )
10 eqcom 2090 . . . . 5  |-  ( { A }  =  { A ,  B }  <->  { A ,  B }  =  { A } )
111, 2, 1preqsn 3617 . . . . 5  |-  ( { A ,  B }  =  { A }  <->  ( A  =  B  /\  B  =  A ) )
12 eqcom 2090 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
1312anbi2i 445 . . . . . 6  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
14 anidm 388 . . . . . 6  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
1513, 14bitri 182 . . . . 5  |-  ( ( A  =  B  /\  B  =  A )  <->  A  =  B )
1610, 11, 153bitri 204 . . . 4  |-  ( { A }  =  { A ,  B }  <->  A  =  B )
1716anbi1i 446 . . 3  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  { A ,  B }  =  C ) )
18 dfsn2 3458 . . . . . . 7  |-  { A }  =  { A ,  A }
19 preq2 3518 . . . . . . 7  |-  ( A  =  B  ->  { A ,  A }  =  { A ,  B }
)
2018, 19syl5req 2133 . . . . . 6  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2120eqeq1d 2096 . . . . 5  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  { A }  =  C ) )
22 eqcom 2090 . . . . 5  |-  ( { A }  =  C  <-> 
C  =  { A } )
2321, 22syl6bb 194 . . . 4  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  C  =  { A } ) )
2423pm5.32i 442 . . 3  |-  ( ( A  =  B  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
2517, 24bitri 182 . 2  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
264, 9, 253bitri 204 1  |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3444   {cpr 3445   <.cop 3447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453
This theorem is referenced by:  relop  4582
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