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Theorem opthreg 4573
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4554 (via the preleq 4572 step). See df-op 3616 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
Hypotheses
Ref Expression
preleq.1  |-  A  e. 
_V
preleq.2  |-  B  e. 
_V
preleq.3  |-  C  e. 
_V
preleq.4  |-  D  e. 
_V
Assertion
Ref Expression
opthreg  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem opthreg
StepHypRef Expression
1 preleq.1 . . . . 5  |-  A  e. 
_V
21prid1 3713 . . . 4  |-  A  e. 
{ A ,  B }
3 preleq.3 . . . . 5  |-  C  e. 
_V
43prid1 3713 . . . 4  |-  C  e. 
{ C ,  D }
5 preleq.2 . . . . . 6  |-  B  e. 
_V
6 prexg 4229 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
71, 5, 6mp2an 426 . . . . 5  |-  { A ,  B }  e.  _V
8 preleq.4 . . . . . 6  |-  D  e. 
_V
9 prexg 4229 . . . . . 6  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  { C ,  D }  e.  _V )
103, 8, 9mp2an 426 . . . . 5  |-  { C ,  D }  e.  _V
111, 7, 3, 10preleq 4572 . . . 4  |-  ( ( ( A  e.  { A ,  B }  /\  C  e.  { C ,  D } )  /\  { A ,  { A ,  B } }  =  { C ,  { C ,  D } } )  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D }
) )
122, 4, 11mpanl12 436 . . 3  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D } ) )
13 preq1 3684 . . . . . 6  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
1413eqeq1d 2198 . . . . 5  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D } 
<->  { C ,  B }  =  { C ,  D } ) )
155, 8preqr2 3784 . . . . 5  |-  ( { C ,  B }  =  { C ,  D }  ->  B  =  D )
1614, 15biimtrdi 163 . . . 4  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D }  ->  B  =  D ) )
1716imdistani 445 . . 3  |-  ( ( A  =  C  /\  { A ,  B }  =  { C ,  D } )  ->  ( A  =  C  /\  B  =  D )
)
1812, 17syl 14 . 2  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  ->  ( A  =  C  /\  B  =  D )
)
19 preq1 3684 . . . 4  |-  ( A  =  C  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
2019adantr 276 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
21 preq12 3686 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
2221preq2d 3691 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { C ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2320, 22eqtrd 2222 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2418, 23impbii 126 1  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   _Vcvv 2752   {cpr 3608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pr 4227  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-sn 3613  df-pr 3614
This theorem is referenced by: (None)
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