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Theorem opthreg 6798
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6785 (via the preleq 6797 step). See df-op 3262 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 3342 . . . 4  |-  A  e. 
{ A ,  B }
3 preleq.3 . . . . 5  |-  C  e. 
_V
43prid1 3342 . . . 4  |-  C  e. 
{ C ,  D }
5 prex 3770 . . . . 5  |-  { A ,  B }  e.  _V
6 prex 3770 . . . . 5  |-  { C ,  D }  e.  _V
71, 5, 3, 6preleq 6797 . . . 4  |-  ( ( ( A  e.  { A ,  B }  /\  C  e.  { C ,  D } )  /\  { A ,  { A ,  B } }  =  { C ,  { C ,  D } } )  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D }
) )
82, 4, 7mpanl12 655 . . 3  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D } ) )
9 preq1 3314 . . . . . 6  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
109eqeq1d 2070 . . . . 5  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D } 
<->  { C ,  B }  =  { C ,  D } ) )
11 preleq.2 . . . . . 6  |-  B  e. 
_V
12 preleq.4 . . . . . 6  |-  D  e. 
_V
1311, 12preqr2 3387 . . . . 5  |-  ( { C ,  B }  =  { C ,  D }  ->  B  =  D )
1410, 13syl6bi 217 . . . 4  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D }  ->  B  =  D ) )
1514imdistani 663 . . 3  |-  ( ( A  =  C  /\  { A ,  B }  =  { C ,  D } )  ->  ( A  =  C  /\  B  =  D )
)
168, 15syl 15 . 2  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  ->  ( A  =  C  /\  B  =  D )
)
17 preq1 3314 . . . 4  |-  ( A  =  C  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
1817adantr 444 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
19 preq12 3316 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
2019preq2d 3321 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { C ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2118, 20eqtrd 2094 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2216, 21impbii 178 1  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 174    /\ wa 356    = wceq 1518    e. wcel 1520   _Vcvv 2473   {cpr 3254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1440  ax-6 1441  ax-7 1442  ax-gen 1443  ax-8 1522  ax-11 1523  ax-14 1525  ax-17 1527  ax-12o 1560  ax-10 1574  ax-9 1580  ax-4 1587  ax-16 1773  ax-ext 2044  ax-sep 3699  ax-nul 3707  ax-pr 3767  ax-reg 6785
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 896  df-ex 1445  df-sb 1734  df-eu 1956  df-mo 1957  df-clab 2050  df-cleq 2055  df-clel 2058  df-ne 2182  df-ral 2276  df-rex 2277  df-rab 2279  df-v 2475  df-sbc 2649  df-dif 2794  df-un 2796  df-in 2798  df-ss 2802  df-nul 3071  df-if 3180  df-sn 3259  df-pr 3260  df-op 3262  df-br 3585  df-opab 3639  df-eprel 3854  df-fr 3901
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