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Theorem disjen 7729
Description: A stronger form of pwuninel 7022. We can use pwuninel 7022, 2pwuninel 7727 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 6830 . . . . . . . 8  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
21ad2antll 735 . . . . . . 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 764 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  ->  x  e.  A )
42, 3eqeltrrd 2530 . . . . . 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 5875 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6 fvex 5875 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
75, 6opelrn 5066 . . . . . 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 7022 . . . . . 6  |-  -.  ~P U.
ran  A  e.  ran  A
10 xp2nd 6824 . . . . . . . . 9  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  ( 2nd `  x )  e. 
{ ~P U. ran  A } )
1110ad2antll 735 . . . . . . . 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 3993 . . . . . . . 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 2513 . . . . . 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 305 . . . . 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 580 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )
17 elin 3617 . . . 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 307 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  x  e.  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
1918eq0rdv 3769 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  i^i  ( B  X.  { ~P U. ran  A } ) )  =  (/) )
20 simpr 463 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
21 rnexg 6725 . . . . 5  |-  ( A  e.  V  ->  ran  A  e.  _V )
2221adantr 467 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  A  e.  _V )
23 uniexg 6588 . . . 4  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
24 pwexg 4587 . . . 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 7657 . . 3  |-  ( ( B  e.  W  /\  ~P U. ran  A  e. 
_V )  ->  ( B  X.  { ~P U. ran  A } )  ~~  B )
2720, 25, 26syl2anc 667 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  ~~  B )
2819, 27jca 535 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 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    i^i cin 3403   (/)c0 3731   ~Pcpw 3951   {csn 3968   <.cop 3974   U.cuni 4198   class class class wbr 4402    X. cxp 4832   ran crn 4835   ` cfv 5582   1stc1st 6791   2ndc2nd 6792    ~~ cen 7566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-1st 6793  df-2nd 6794  df-en 7570
This theorem is referenced by:  disjenex  7730  domss2  7731
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