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Theorem disjen 7747
Description: A stronger form of pwuninel 7040. We can use pwuninel 7040, 2pwuninel 7745 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 6849 . . . . . . . 8  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
21ad2antll 743 . . . . . . 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 772 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  ->  x  e.  A )
42, 3eqeltrrd 2550 . . . . . 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 5889 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6 fvex 5889 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
75, 6opelrn 5072 . . . . . 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 7040 . . . . . 6  |-  -.  ~P U.
ran  A  e.  ran  A
10 xp2nd 6843 . . . . . . . . 9  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  ( 2nd `  x )  e. 
{ ~P U. ran  A } )
1110ad2antll 743 . . . . . . . 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 3985 . . . . . . . 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 2533 . . . . . 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 310 . . . . 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 586 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )
17 elin 3608 . . . 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 312 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  x  e.  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
1918eq0rdv 3773 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  i^i  ( B  X.  { ~P U. ran  A } ) )  =  (/) )
20 simpr 468 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
21 rnexg 6744 . . . . 5  |-  ( A  e.  V  ->  ran  A  e.  _V )
2221adantr 472 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  A  e.  _V )
23 uniexg 6607 . . . 4  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
24 pwexg 4585 . . . 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 7675 . . 3  |-  ( ( B  e.  W  /\  ~P U. ran  A  e. 
_V )  ->  ( B  X.  { ~P U. ran  A } )  ~~  B )
2720, 25, 26syl2anc 673 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  ~~  B )
2819, 27jca 541 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 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389   (/)c0 3722   ~Pcpw 3942   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395    X. cxp 4837   ran crn 4840   ` cfv 5589   1stc1st 6810   2ndc2nd 6811    ~~ cen 7584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-1st 6812  df-2nd 6813  df-en 7588
This theorem is referenced by:  disjenex  7748  domss2  7749
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