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Theorem opthreg 4538
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4519 (via the preleq 4537 step). See df-op 3590 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 3687 . . . 4  |-  A  e. 
{ A ,  B }
3 preleq.3 . . . . 5  |-  C  e. 
_V
43prid1 3687 . . . 4  |-  C  e. 
{ C ,  D }
5 preleq.2 . . . . . 6  |-  B  e. 
_V
6 prexg 4194 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
71, 5, 6mp2an 424 . . . . 5  |-  { A ,  B }  e.  _V
8 preleq.4 . . . . . 6  |-  D  e. 
_V
9 prexg 4194 . . . . . 6  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  { C ,  D }  e.  _V )
103, 8, 9mp2an 424 . . . . 5  |-  { C ,  D }  e.  _V
111, 7, 3, 10preleq 4537 . . . 4  |-  ( ( ( A  e.  { A ,  B }  /\  C  e.  { C ,  D } )  /\  { A ,  { A ,  B } }  =  { C ,  { C ,  D } } )  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D }
) )
122, 4, 11mpanl12 434 . . 3  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D } ) )
13 preq1 3658 . . . . . 6  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
1413eqeq1d 2179 . . . . 5  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D } 
<->  { C ,  B }  =  { C ,  D } ) )
155, 8preqr2 3754 . . . . 5  |-  ( { C ,  B }  =  { C ,  D }  ->  B  =  D )
1614, 15syl6bi 162 . . . 4  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D }  ->  B  =  D ) )
1716imdistani 443 . . 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 3658 . . . 4  |-  ( A  =  C  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
2019adantr 274 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
21 preq12 3660 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
2221preq2d 3665 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { C ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2320, 22eqtrd 2203 . 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 1348    e. wcel 2141   _Vcvv 2730   {cpr 3582
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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pr 4192  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-sn 3587  df-pr 3588
This theorem is referenced by: (None)
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