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Theorem opthreg 4409
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4390 (via the preleq 4408 step). See df-op 3483 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 3576 . . . 4  |-  A  e. 
{ A ,  B }
3 preleq.3 . . . . 5  |-  C  e. 
_V
43prid1 3576 . . . 4  |-  C  e. 
{ C ,  D }
5 preleq.2 . . . . . 6  |-  B  e. 
_V
6 prexg 4071 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
71, 5, 6mp2an 420 . . . . 5  |-  { A ,  B }  e.  _V
8 preleq.4 . . . . . 6  |-  D  e. 
_V
9 prexg 4071 . . . . . 6  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  { C ,  D }  e.  _V )
103, 8, 9mp2an 420 . . . . 5  |-  { C ,  D }  e.  _V
111, 7, 3, 10preleq 4408 . . . 4  |-  ( ( ( A  e.  { A ,  B }  /\  C  e.  { C ,  D } )  /\  { A ,  { A ,  B } }  =  { C ,  { C ,  D } } )  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D }
) )
122, 4, 11mpanl12 430 . . 3  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D } ) )
13 preq1 3547 . . . . . 6  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
1413eqeq1d 2108 . . . . 5  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D } 
<->  { C ,  B }  =  { C ,  D } ) )
155, 8preqr2 3643 . . . . 5  |-  ( { C ,  B }  =  { C ,  D }  ->  B  =  D )
1614, 15syl6bi 162 . . . 4  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D }  ->  B  =  D ) )
1716imdistani 437 . . 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 3547 . . . 4  |-  ( A  =  C  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
2019adantr 272 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
21 preq12 3549 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
2221preq2d 3554 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { C ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2320, 22eqtrd 2132 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2418, 23impbii 125 1  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   _Vcvv 2641   {cpr 3475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pr 4069  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-dif 3023  df-un 3025  df-sn 3480  df-pr 3481
This theorem is referenced by: (None)
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