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Theorem disjen 7468
Description: A stronger form of pwuninel 6794. We can use pwuninel 6794, 2pwuninel 7466 to create one or two sets disjoint from a given set  A, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set  B we can construct a set  x that is equinumerous to it and disjoint from  A. (Contributed by Mario Carneiro, 7-Feb-2015.)
Assertion
Ref Expression
disjen  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  i^i  ( B  X.  { ~P U.
ran  A } ) )  =  (/)  /\  ( B  X.  { ~P U. ran  A } )  ~~  B ) )

Proof of Theorem disjen
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 1st2nd2 6613 . . . . . . . 8  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
21ad2antll 728 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  ->  x  =  <. ( 1st `  x ) ,  ( 2nd `  x )
>. )
3 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  ->  x  e.  A )
42, 3eqeltrrd 2518 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  ->  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  A
)
5 fvex 5701 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6 fvex 5701 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
75, 6opelrn 5071 . . . . . 6  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  A  ->  ( 2nd `  x
)  e.  ran  A
)
84, 7syl 16 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  -> 
( 2nd `  x
)  e.  ran  A
)
9 pwuninel 6794 . . . . . 6  |-  -.  ~P U.
ran  A  e.  ran  A
10 xp2nd 6607 . . . . . . . . 9  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  ( 2nd `  x )  e. 
{ ~P U. ran  A } )
1110ad2antll 728 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  -> 
( 2nd `  x
)  e.  { ~P U.
ran  A } )
12 elsni 3902 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  { ~P U. ran  A }  ->  ( 2nd `  x )  =  ~P U. ran  A
)
1311, 12syl 16 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  -> 
( 2nd `  x
)  =  ~P U. ran  A )
1413eleq1d 2509 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  -> 
( ( 2nd `  x
)  e.  ran  A  <->  ~P
U. ran  A  e.  ran  A ) )
159, 14mtbiri 303 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  ->  -.  ( 2nd `  x
)  e.  ran  A
)
168, 15pm2.65da 576 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )
17 elin 3539 . . . 4  |-  ( x  e.  ( A  i^i  ( B  X.  { ~P U.
ran  A } ) )  <->  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )
1816, 17sylnibr 305 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  x  e.  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
1918eq0rdv 3672 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  i^i  ( B  X.  { ~P U. ran  A } ) )  =  (/) )
20 simpr 461 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
21 rnexg 6510 . . . . 5  |-  ( A  e.  V  ->  ran  A  e.  _V )
2221adantr 465 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  A  e.  _V )
23 uniexg 6377 . . . 4  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
24 pwexg 4476 . . . 4  |-  ( U. ran  A  e.  _V  ->  ~P
U. ran  A  e.  _V )
2522, 23, 243syl 20 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P U. ran  A  e.  _V )
26 xpsneng 7396 . . 3  |-  ( ( B  e.  W  /\  ~P U. ran  A  e. 
_V )  ->  ( B  X.  { ~P U. ran  A } )  ~~  B )
2720, 25, 26syl2anc 661 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  ~~  B )
2819, 27jca 532 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  i^i  ( B  X.  { ~P U.
ran  A } ) )  =  (/)  /\  ( B  X.  { ~P U. ran  A } )  ~~  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972    i^i cin 3327   (/)c0 3637   ~Pcpw 3860   {csn 3877   <.cop 3883   U.cuni 4091   class class class wbr 4292    X. cxp 4838   ran crn 4841   ` cfv 5418   1stc1st 6575   2ndc2nd 6576    ~~ cen 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-1st 6577  df-2nd 6578  df-en 7311
This theorem is referenced by:  disjenex  7469  domss2  7470
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