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Theorem opthprc 4736
Description: Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)
Assertion
Ref Expression
opthprc  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem opthprc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2344 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  ( <. x ,  (/) >.  e.  (
( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  <->  <. x ,  (/) >.  e.  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) ) ) )
2 0ex 4150 . . . . . . . . 9  |-  (/)  e.  _V
32snid 3667 . . . . . . . 8  |-  (/)  e.  { (/)
}
4 opelxp 4719 . . . . . . . 8  |-  ( <.
x ,  (/) >.  e.  ( A  X.  { (/) } )  <->  ( x  e.  A  /\  (/)  e.  { (/)
} ) )
53, 4mpbiran2 885 . . . . . . 7  |-  ( <.
x ,  (/) >.  e.  ( A  X.  { (/) } )  <->  x  e.  A
)
6 opelxp 4719 . . . . . . . 8  |-  ( <.
x ,  (/) >.  e.  ( B  X.  { { (/)
} } )  <->  ( x  e.  B  /\  (/)  e.  { { (/) } } ) )
7 0nep0 4181 . . . . . . . . . 10  |-  (/)  =/=  { (/)
}
82elsnc 3663 . . . . . . . . . 10  |-  ( (/)  e.  { { (/) } }  <->  (/)  =  { (/) } )
97, 8nemtbir 2534 . . . . . . . . 9  |-  -.  (/)  e.  { { (/) } }
109bianfi 891 . . . . . . . 8  |-  ( (/)  e.  { { (/) } }  <->  ( x  e.  B  /\  (/) 
e.  { { (/) } } ) )
116, 10bitr4i 243 . . . . . . 7  |-  ( <.
x ,  (/) >.  e.  ( B  X.  { { (/)
} } )  <->  (/)  e.  { { (/) } } )
125, 11orbi12i 507 . . . . . 6  |-  ( (
<. x ,  (/) >.  e.  ( A  X.  { (/) } )  \/  <. x ,  (/) >.  e.  ( B  X.  { { (/) } } ) )  <->  ( x  e.  A  \/  (/)  e.  { { (/) } } ) )
13 elun 3316 . . . . . 6  |-  ( <.
x ,  (/) >.  e.  ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  <->  ( <. x ,  (/) >.  e.  ( A  X.  { (/) } )  \/  <. x ,  (/) >.  e.  ( B  X.  { { (/) } } ) ) )
149biorfi 396 . . . . . 6  |-  ( x  e.  A  <->  ( x  e.  A  \/  (/)  e.  { { (/) } } ) )
1512, 13, 143bitr4ri 269 . . . . 5  |-  ( x  e.  A  <->  <. x ,  (/) >.  e.  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) ) )
16 opelxp 4719 . . . . . . . 8  |-  ( <.
x ,  (/) >.  e.  ( C  X.  { (/) } )  <->  ( x  e.  C  /\  (/)  e.  { (/)
} ) )
173, 16mpbiran2 885 . . . . . . 7  |-  ( <.
x ,  (/) >.  e.  ( C  X.  { (/) } )  <->  x  e.  C
)
18 opelxp 4719 . . . . . . . 8  |-  ( <.
x ,  (/) >.  e.  ( D  X.  { { (/)
} } )  <->  ( x  e.  D  /\  (/)  e.  { { (/) } } ) )
199bianfi 891 . . . . . . . 8  |-  ( (/)  e.  { { (/) } }  <->  ( x  e.  D  /\  (/) 
e.  { { (/) } } ) )
2018, 19bitr4i 243 . . . . . . 7  |-  ( <.
x ,  (/) >.  e.  ( D  X.  { { (/)
} } )  <->  (/)  e.  { { (/) } } )
2117, 20orbi12i 507 . . . . . 6  |-  ( (
<. x ,  (/) >.  e.  ( C  X.  { (/) } )  \/  <. x ,  (/) >.  e.  ( D  X.  { { (/) } } ) )  <->  ( x  e.  C  \/  (/)  e.  { { (/) } } ) )
22 elun 3316 . . . . . 6  |-  ( <.
x ,  (/) >.  e.  ( ( C  X.  { (/)
} )  u.  ( D  X.  { { (/) } } ) )  <->  ( <. x ,  (/) >.  e.  ( C  X.  { (/) } )  \/  <. x ,  (/) >.  e.  ( D  X.  { { (/) } } ) ) )
239biorfi 396 . . . . . 6  |-  ( x  e.  C  <->  ( x  e.  C  \/  (/)  e.  { { (/) } } ) )
2421, 22, 233bitr4ri 269 . . . . 5  |-  ( x  e.  C  <->  <. x ,  (/) >.  e.  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) ) )
251, 15, 243bitr4g 279 . . . 4  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  ( x  e.  A  <->  x  e.  C
) )
2625eqrdv 2281 . . 3  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  A  =  C )
27 eleq2 2344 . . . . 5  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  ( <. x ,  { (/) } >.  e.  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) )  <->  <. x ,  { (/)
} >.  e.  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) ) ) )
28 opelxp 4719 . . . . . . . 8  |-  ( <.
x ,  { (/) }
>.  e.  ( A  X.  { (/) } )  <->  ( x  e.  A  /\  { (/) }  e.  { (/) } ) )
29 p0ex 4197 . . . . . . . . . . . 12  |-  { (/) }  e.  _V
3029elsnc 3663 . . . . . . . . . . 11  |-  ( {
(/) }  e.  { (/) }  <->  { (/) }  =  (/) )
31 eqcom 2285 . . . . . . . . . . 11  |-  ( {
(/) }  =  (/)  <->  (/)  =  { (/)
} )
3230, 31bitri 240 . . . . . . . . . 10  |-  ( {
(/) }  e.  { (/) }  <->  (/)  =  { (/) } )
337, 32nemtbir 2534 . . . . . . . . 9  |-  -.  { (/)
}  e.  { (/) }
3433bianfi 891 . . . . . . . 8  |-  ( {
(/) }  e.  { (/) }  <-> 
( x  e.  A  /\  { (/) }  e.  { (/)
} ) )
3528, 34bitr4i 243 . . . . . . 7  |-  ( <.
x ,  { (/) }
>.  e.  ( A  X.  { (/) } )  <->  { (/) }  e.  {
(/) } )
3629snid 3667 . . . . . . . 8  |-  { (/) }  e.  { { (/) } }
37 opelxp 4719 . . . . . . . 8  |-  ( <.
x ,  { (/) }
>.  e.  ( B  X.  { { (/) } } )  <-> 
( x  e.  B  /\  { (/) }  e.  { { (/) } } ) )
3836, 37mpbiran2 885 . . . . . . 7  |-  ( <.
x ,  { (/) }
>.  e.  ( B  X.  { { (/) } } )  <-> 
x  e.  B )
3935, 38orbi12i 507 . . . . . 6  |-  ( (
<. x ,  { (/) }
>.  e.  ( A  X.  { (/) } )  \/ 
<. x ,  { (/) }
>.  e.  ( B  X.  { { (/) } } ) )  <->  ( { (/) }  e.  { (/) }  \/  x  e.  B )
)
40 elun 3316 . . . . . 6  |-  ( <.
x ,  { (/) }
>.  e.  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) )  <->  ( <. x ,  { (/) } >.  e.  ( A  X.  { (/) } )  \/  <. x ,  { (/) } >.  e.  ( B  X.  { { (/)
} } ) ) )
41 biorf 394 . . . . . . 7  |-  ( -. 
{ (/) }  e.  { (/)
}  ->  ( x  e.  B  <->  ( { (/) }  e.  { (/) }  \/  x  e.  B )
) )
4233, 41ax-mp 8 . . . . . 6  |-  ( x  e.  B  <->  ( { (/)
}  e.  { (/) }  \/  x  e.  B
) )
4339, 40, 423bitr4ri 269 . . . . 5  |-  ( x  e.  B  <->  <. x ,  { (/) } >.  e.  ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) ) )
44 opelxp 4719 . . . . . . . 8  |-  ( <.
x ,  { (/) }
>.  e.  ( C  X.  { (/) } )  <->  ( x  e.  C  /\  { (/) }  e.  { (/) } ) )
4533bianfi 891 . . . . . . . 8  |-  ( {
(/) }  e.  { (/) }  <-> 
( x  e.  C  /\  { (/) }  e.  { (/)
} ) )
4644, 45bitr4i 243 . . . . . . 7  |-  ( <.
x ,  { (/) }
>.  e.  ( C  X.  { (/) } )  <->  { (/) }  e.  {
(/) } )
47 opelxp 4719 . . . . . . . 8  |-  ( <.
x ,  { (/) }
>.  e.  ( D  X.  { { (/) } } )  <-> 
( x  e.  D  /\  { (/) }  e.  { { (/) } } ) )
4836, 47mpbiran2 885 . . . . . . 7  |-  ( <.
x ,  { (/) }
>.  e.  ( D  X.  { { (/) } } )  <-> 
x  e.  D )
4946, 48orbi12i 507 . . . . . 6  |-  ( (
<. x ,  { (/) }
>.  e.  ( C  X.  { (/) } )  \/ 
<. x ,  { (/) }
>.  e.  ( D  X.  { { (/) } } ) )  <->  ( { (/) }  e.  { (/) }  \/  x  e.  D )
)
50 elun 3316 . . . . . 6  |-  ( <.
x ,  { (/) }
>.  e.  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  <->  ( <. x ,  { (/) } >.  e.  ( C  X.  { (/) } )  \/  <. x ,  { (/) } >.  e.  ( D  X.  { { (/)
} } ) ) )
51 biorf 394 . . . . . . 7  |-  ( -. 
{ (/) }  e.  { (/)
}  ->  ( x  e.  D  <->  ( { (/) }  e.  { (/) }  \/  x  e.  D )
) )
5233, 51ax-mp 8 . . . . . 6  |-  ( x  e.  D  <->  ( { (/)
}  e.  { (/) }  \/  x  e.  D
) )
5349, 50, 523bitr4ri 269 . . . . 5  |-  ( x  e.  D  <->  <. x ,  { (/) } >.  e.  ( ( C  X.  { (/)
} )  u.  ( D  X.  { { (/) } } ) ) )
5427, 43, 533bitr4g 279 . . . 4  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  ( x  e.  B  <->  x  e.  D
) )
5554eqrdv 2281 . . 3  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  B  =  D )
5626, 55jca 518 . 2  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  ( A  =  C  /\  B  =  D ) )
57 xpeq1 4703 . . 3  |-  ( A  =  C  ->  ( A  X.  { (/) } )  =  ( C  X.  { (/) } ) )
58 xpeq1 4703 . . 3  |-  ( B  =  D  ->  ( B  X.  { { (/) } } )  =  ( D  X.  { { (/)
} } ) )
59 uneq12 3324 . . 3  |-  ( ( ( A  X.  { (/)
} )  =  ( C  X.  { (/) } )  /\  ( B  X.  { { (/) } } )  =  ( D  X.  { { (/)
} } ) )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) ) )
6057, 58, 59syl2an 463 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) ) )
6156, 60impbii 180 1  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150   (/)c0 3455   {csn 3640   <.cop 3643    X. cxp 4687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
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