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Theorem opthreg 6782
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 6769 (via the preleq 6781 step). See df-op 3281 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 3358 . . . 4  |-  A  e.  { A ,  B }
3 preleq.3 . . . . 5  |-  C  e.  _V
43prid1 3358 . . . 4  |-  C  e.  { C ,  D }
5 prex 3786 . . . . 5  |-  { A ,  B }  e.  _V
6 prex 3786 . . . . 5  |-  { C ,  D }  e.  _V
71, 5, 3, 6preleq 6781 . . . 4  |-  ( (
( A  e.  { A ,  B }  /\  C  e.  { C ,  D } )  /\  { A ,  { A ,  B } }  =  { C ,  { C ,  D } } )  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D }
) )
82, 4, 7mpanl12 659 . . 3  |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  ->  ( A  =  C  /\  { A ,  B }  =  { C ,  D } ) )
9 preq1 3333 . . . . . 6  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
109eqeq1d 2090 . . . . 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 3403 . . . . 5  |-  ( { C ,  B }  =  { C ,  D }  ->  B  =  D )
1410, 13syl6bi 217 . . . 4  |-  ( A  =  C  ->  ( { A ,  B }  =  { C ,  D }  ->  B  =  D ) )
1514imdistani 667 . . 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 3333 . . . 4  |-  ( A  =  C  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
1817adantr 445 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  { A ,  B } }  =  { C ,  { A ,  B } } )
19 preq12 3335 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
2019preq2d 3340 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { C ,  { A ,  B } }  =  { C ,  { C ,  D } } )
2118, 20eqtrd 2114 . 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 357    = wceq 1536    e. wcel 1538   _Vcvv 2492   {cpr 3273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1451  ax-6 1452  ax-7 1453  ax-gen 1454  ax-8 1540  ax-11 1541  ax-14 1543  ax-17 1545  ax-12o 1578  ax-10 1592  ax-9 1598  ax-4 1606  ax-16 1793  ax-ext 2064  ax-sep 3715  ax-nul 3723  ax-pr 3783  ax-reg 6769
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3an 905  df-ex 1456  df-sb 1754  df-eu 1976  df-mo 1977  df-clab 2070  df-cleq 2075  df-clel 2078  df-ne 2201  df-ral 2295  df-rex 2296  df-rab 2298  df-v 2494  df-sbc 2668  df-dif 2813  df-un 2815  df-in 2817  df-ss 2821  df-nul 3089  df-if 3199  df-sn 3278  df-pr 3279  df-op 3281  df-br 3601  df-opab 3655  df-eprel 3870  df-fr 3917
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