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Theorem disjen 7735
Description: A stronger form of pwuninel 7030. We can use pwuninel 7030, 2pwuninel 7733 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 6844 . . . . . . . 8  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
21ad2antll 733 . . . . . . 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 762 . . . . . . 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 5891 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6 fvex 5891 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
75, 6opelrn 5086 . . . . . 6  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  A  ->  ( 2nd `  x
)  e.  ran  A
)
84, 7syl 17 . . . . 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 7030 . . . . . 6  |-  -.  ~P U.
ran  A  e.  ran  A
10 xp2nd 6838 . . . . . . . . 9  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  ( 2nd `  x )  e. 
{ ~P U. ran  A } )
1110ad2antll 733 . . . . . . . 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 4027 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  { ~P U. ran  A }  ->  ( 2nd `  x )  =  ~P U. ran  A
)
1311, 12syl 17 . . . . . . 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 2498 . . . . . 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 304 . . . . 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 578 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )
17 elin 3655 . . . 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 306 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  x  e.  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
1918eq0rdv 3803 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  i^i  ( B  X.  { ~P U. ran  A } ) )  =  (/) )
20 simpr 462 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
21 rnexg 6739 . . . . 5  |-  ( A  e.  V  ->  ran  A  e.  _V )
2221adantr 466 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  A  e.  _V )
23 uniexg 6602 . . . 4  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
24 pwexg 4609 . . . 4  |-  ( U. ran  A  e.  _V  ->  ~P
U. ran  A  e.  _V )
2522, 23, 243syl 18 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P U. ran  A  e.  _V )
26 xpsneng 7663 . . 3  |-  ( ( B  e.  W  /\  ~P U. ran  A  e. 
_V )  ->  ( B  X.  { ~P U. ran  A } )  ~~  B )
2720, 25, 26syl2anc 665 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  ~~  B )
2819, 27jca 534 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 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    i^i cin 3441   (/)c0 3767   ~Pcpw 3985   {csn 4002   <.cop 4008   U.cuni 4222   class class class wbr 4426    X. cxp 4852   ran crn 4855   ` cfv 5601   1stc1st 6805   2ndc2nd 6806    ~~ cen 7574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1st 6807  df-2nd 6808  df-en 7578
This theorem is referenced by:  disjenex  7736  domss2  7737
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