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Theorem opthreg 4592
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4573 (via the preleq 4591 step). See df-op 3631 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 3728 . . . 4  |-  A  e. 
{ A ,  B }
3 preleq.3 . . . . 5  |-  C  e. 
_V
43prid1 3728 . . . 4  |-  C  e. 
{ C ,  D }
5 preleq.2 . . . . . 6  |-  B  e. 
_V
6 prexg 4244 . . . . . 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 4244 . . . . . 6  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  { C ,  D }  e.  _V )
103, 8, 9mp2an 426 . . . . 5  |-  { C ,  D }  e.  _V
111, 7, 3, 10preleq 4591 . . . 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 3699 . . . . . 6  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
1413eqeq1d 2205 . . . . 5  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D } 
<->  { C ,  B }  =  { C ,  D } ) )
155, 8preqr2 3799 . . . . 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 3699 . . . 4  |-  ( A  =  C  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
2019adantr 276 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
21 preq12 3701 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
2221preq2d 3706 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { C ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2320, 22eqtrd 2229 . 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 2167   _Vcvv 2763   {cpr 3623
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pr 4242  ax-setind 4573
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-un 3161  df-sn 3628  df-pr 3629
This theorem is referenced by: (None)
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