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Theorem opthprc 4884
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 2465 . . . . 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 4299 . . . . . . . . 9  |-  (/)  e.  _V
32snid 3801 . . . . . . . 8  |-  (/)  e.  { (/)
}
4 opelxp 4867 . . . . . . . 8  |-  ( <.
x ,  (/) >.  e.  ( A  X.  { (/) } )  <->  ( x  e.  A  /\  (/)  e.  { (/)
} ) )
53, 4mpbiran2 886 . . . . . . 7  |-  ( <.
x ,  (/) >.  e.  ( A  X.  { (/) } )  <->  x  e.  A
)
6 opelxp 4867 . . . . . . . 8  |-  ( <.
x ,  (/) >.  e.  ( B  X.  { { (/)
} } )  <->  ( x  e.  B  /\  (/)  e.  { { (/) } } ) )
7 0nep0 4330 . . . . . . . . . 10  |-  (/)  =/=  { (/)
}
82elsnc 3797 . . . . . . . . . 10  |-  ( (/)  e.  { { (/) } }  <->  (/)  =  { (/) } )
97, 8nemtbir 2655 . . . . . . . . 9  |-  -.  (/)  e.  { { (/) } }
109bianfi 892 . . . . . . . 8  |-  ( (/)  e.  { { (/) } }  <->  ( x  e.  B  /\  (/) 
e.  { { (/) } } ) )
116, 10bitr4i 244 . . . . . . 7  |-  ( <.
x ,  (/) >.  e.  ( B  X.  { { (/)
} } )  <->  (/)  e.  { { (/) } } )
125, 11orbi12i 508 . . . . . 6  |-  ( (
<. x ,  (/) >.  e.  ( A  X.  { (/) } )  \/  <. x ,  (/) >.  e.  ( B  X.  { { (/) } } ) )  <->  ( x  e.  A  \/  (/)  e.  { { (/) } } ) )
13 elun 3448 . . . . . 6  |-  ( <.
x ,  (/) >.  e.  ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  <->  ( <. x ,  (/) >.  e.  ( A  X.  { (/) } )  \/  <. x ,  (/) >.  e.  ( B  X.  { { (/) } } ) ) )
149biorfi 397 . . . . . 6  |-  ( x  e.  A  <->  ( x  e.  A  \/  (/)  e.  { { (/) } } ) )
1512, 13, 143bitr4ri 270 . . . . 5  |-  ( x  e.  A  <->  <. x ,  (/) >.  e.  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) ) )
16 opelxp 4867 . . . . . . . 8  |-  ( <.
x ,  (/) >.  e.  ( C  X.  { (/) } )  <->  ( x  e.  C  /\  (/)  e.  { (/)
} ) )
173, 16mpbiran2 886 . . . . . . 7  |-  ( <.
x ,  (/) >.  e.  ( C  X.  { (/) } )  <->  x  e.  C
)
18 opelxp 4867 . . . . . . . 8  |-  ( <.
x ,  (/) >.  e.  ( D  X.  { { (/)
} } )  <->  ( x  e.  D  /\  (/)  e.  { { (/) } } ) )
199bianfi 892 . . . . . . . 8  |-  ( (/)  e.  { { (/) } }  <->  ( x  e.  D  /\  (/) 
e.  { { (/) } } ) )
2018, 19bitr4i 244 . . . . . . 7  |-  ( <.
x ,  (/) >.  e.  ( D  X.  { { (/)
} } )  <->  (/)  e.  { { (/) } } )
2117, 20orbi12i 508 . . . . . 6  |-  ( (
<. x ,  (/) >.  e.  ( C  X.  { (/) } )  \/  <. x ,  (/) >.  e.  ( D  X.  { { (/) } } ) )  <->  ( x  e.  C  \/  (/)  e.  { { (/) } } ) )
22 elun 3448 . . . . . 6  |-  ( <.
x ,  (/) >.  e.  ( ( C  X.  { (/)
} )  u.  ( D  X.  { { (/) } } ) )  <->  ( <. x ,  (/) >.  e.  ( C  X.  { (/) } )  \/  <. x ,  (/) >.  e.  ( D  X.  { { (/) } } ) ) )
239biorfi 397 . . . . . 6  |-  ( x  e.  C  <->  ( x  e.  C  \/  (/)  e.  { { (/) } } ) )
2421, 22, 233bitr4ri 270 . . . . 5  |-  ( x  e.  C  <->  <. x ,  (/) >.  e.  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) ) )
251, 15, 243bitr4g 280 . . . 4  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  ( x  e.  A  <->  x  e.  C
) )
2625eqrdv 2402 . . 3  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  A  =  C )
27 eleq2 2465 . . . . 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 4867 . . . . . . . 8  |-  ( <.
x ,  { (/) }
>.  e.  ( A  X.  { (/) } )  <->  ( x  e.  A  /\  { (/) }  e.  { (/) } ) )
29 p0ex 4346 . . . . . . . . . . . 12  |-  { (/) }  e.  _V
3029elsnc 3797 . . . . . . . . . . 11  |-  ( {
(/) }  e.  { (/) }  <->  { (/) }  =  (/) )
31 eqcom 2406 . . . . . . . . . . 11  |-  ( {
(/) }  =  (/)  <->  (/)  =  { (/)
} )
3230, 31bitri 241 . . . . . . . . . 10  |-  ( {
(/) }  e.  { (/) }  <->  (/)  =  { (/) } )
337, 32nemtbir 2655 . . . . . . . . 9  |-  -.  { (/)
}  e.  { (/) }
3433bianfi 892 . . . . . . . 8  |-  ( {
(/) }  e.  { (/) }  <-> 
( x  e.  A  /\  { (/) }  e.  { (/)
} ) )
3528, 34bitr4i 244 . . . . . . 7  |-  ( <.
x ,  { (/) }
>.  e.  ( A  X.  { (/) } )  <->  { (/) }  e.  {
(/) } )
3629snid 3801 . . . . . . . 8  |-  { (/) }  e.  { { (/) } }
37 opelxp 4867 . . . . . . . 8  |-  ( <.
x ,  { (/) }
>.  e.  ( B  X.  { { (/) } } )  <-> 
( x  e.  B  /\  { (/) }  e.  { { (/) } } ) )
3836, 37mpbiran2 886 . . . . . . 7  |-  ( <.
x ,  { (/) }
>.  e.  ( B  X.  { { (/) } } )  <-> 
x  e.  B )
3935, 38orbi12i 508 . . . . . 6  |-  ( (
<. x ,  { (/) }
>.  e.  ( A  X.  { (/) } )  \/ 
<. x ,  { (/) }
>.  e.  ( B  X.  { { (/) } } ) )  <->  ( { (/) }  e.  { (/) }  \/  x  e.  B )
)
40 elun 3448 . . . . . 6  |-  ( <.
x ,  { (/) }
>.  e.  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) )  <->  ( <. x ,  { (/) } >.  e.  ( A  X.  { (/) } )  \/  <. x ,  { (/) } >.  e.  ( B  X.  { { (/)
} } ) ) )
41 biorf 395 . . . . . . 7  |-  ( -. 
{ (/) }  e.  { (/)
}  ->  ( x  e.  B  <->  ( { (/) }  e.  { (/) }  \/  x  e.  B )
) )
4233, 41ax-mp 8 . . . . . 6  |-  ( x  e.  B  <->  ( { (/)
}  e.  { (/) }  \/  x  e.  B
) )
4339, 40, 423bitr4ri 270 . . . . 5  |-  ( x  e.  B  <->  <. x ,  { (/) } >.  e.  ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) ) )
44 opelxp 4867 . . . . . . . 8  |-  ( <.
x ,  { (/) }
>.  e.  ( C  X.  { (/) } )  <->  ( x  e.  C  /\  { (/) }  e.  { (/) } ) )
4533bianfi 892 . . . . . . . 8  |-  ( {
(/) }  e.  { (/) }  <-> 
( x  e.  C  /\  { (/) }  e.  { (/)
} ) )
4644, 45bitr4i 244 . . . . . . 7  |-  ( <.
x ,  { (/) }
>.  e.  ( C  X.  { (/) } )  <->  { (/) }  e.  {
(/) } )
47 opelxp 4867 . . . . . . . 8  |-  ( <.
x ,  { (/) }
>.  e.  ( D  X.  { { (/) } } )  <-> 
( x  e.  D  /\  { (/) }  e.  { { (/) } } ) )
4836, 47mpbiran2 886 . . . . . . 7  |-  ( <.
x ,  { (/) }
>.  e.  ( D  X.  { { (/) } } )  <-> 
x  e.  D )
4946, 48orbi12i 508 . . . . . 6  |-  ( (
<. x ,  { (/) }
>.  e.  ( C  X.  { (/) } )  \/ 
<. x ,  { (/) }
>.  e.  ( D  X.  { { (/) } } ) )  <->  ( { (/) }  e.  { (/) }  \/  x  e.  D )
)
50 elun 3448 . . . . . 6  |-  ( <.
x ,  { (/) }
>.  e.  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  <->  ( <. x ,  { (/) } >.  e.  ( C  X.  { (/) } )  \/  <. x ,  { (/) } >.  e.  ( D  X.  { { (/)
} } ) ) )
51 biorf 395 . . . . . . 7  |-  ( -. 
{ (/) }  e.  { (/)
}  ->  ( x  e.  D  <->  ( { (/) }  e.  { (/) }  \/  x  e.  D )
) )
5233, 51ax-mp 8 . . . . . 6  |-  ( x  e.  D  <->  ( { (/)
}  e.  { (/) }  \/  x  e.  D
) )
5349, 50, 523bitr4ri 270 . . . . 5  |-  ( x  e.  D  <->  <. x ,  { (/) } >.  e.  ( ( C  X.  { (/)
} )  u.  ( D  X.  { { (/) } } ) ) )
5427, 43, 533bitr4g 280 . . . 4  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  ( x  e.  B  <->  x  e.  D
) )
5554eqrdv 2402 . . 3  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  B  =  D )
5626, 55jca 519 . 2  |-  ( ( ( A  X.  { (/)
} )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) )  ->  ( A  =  C  /\  B  =  D ) )
57 xpeq1 4851 . . 3  |-  ( A  =  C  ->  ( A  X.  { (/) } )  =  ( C  X.  { (/) } ) )
58 xpeq1 4851 . . 3  |-  ( B  =  D  ->  ( B  X.  { { (/) } } )  =  ( D  X.  { { (/)
} } ) )
59 uneq12 3456 . . 3  |-  ( ( ( A  X.  { (/)
} )  =  ( C  X.  { (/) } )  /\  ( B  X.  { { (/) } } )  =  ( D  X.  { { (/)
} } ) )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) ) )
6057, 58, 59syl2an 464 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } ) ) )
6156, 60impbii 181 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    u. cun 3278   (/)c0 3588   {csn 3774   <.cop 3777    X. cxp 4835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-xp 4843
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