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Theorem disjen 7264
Description: A stronger form of pwuninel 6545. We can use pwuninel 6545, 2pwuninel 7262 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 6386 . . . . . . . 8  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
21ad2antll 710 . . . . . . 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 733 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )  ->  x  e.  A )
42, 3eqeltrrd 2511 . . . . . 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 5742 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6 fvex 5742 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
75, 6opelrn 5101 . . . . . 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 6545 . . . . . 6  |-  -.  ~P U.
ran  A  e.  ran  A
10 xp2nd 6377 . . . . . . . . 9  |-  ( x  e.  ( B  X.  { ~P U. ran  A } )  ->  ( 2nd `  x )  e. 
{ ~P U. ran  A } )
1110ad2antll 710 . . . . . . . 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 3838 . . . . . . . 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 2502 . . . . . 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 295 . . . . 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 560 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  ( x  e.  A  /\  x  e.  ( B  X.  { ~P U. ran  A }
) ) )
17 elin 3530 . . . 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 297 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  -.  x  e.  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
1918eq0rdv 3662 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  i^i  ( B  X.  { ~P U. ran  A } ) )  =  (/) )
20 simpr 448 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
21 rnexg 5131 . . . . 5  |-  ( A  e.  V  ->  ran  A  e.  _V )
2221adantr 452 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  A  e.  _V )
23 uniexg 4706 . . . 4  |-  ( ran 
A  e.  _V  ->  U.
ran  A  e.  _V )
24 pwexg 4383 . . . 4  |-  ( U. ran  A  e.  _V  ->  ~P
U. ran  A  e.  _V )
2522, 23, 243syl 19 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P U. ran  A  e.  _V )
26 xpsneng 7193 . . 3  |-  ( ( B  e.  W  /\  ~P U. ran  A  e. 
_V )  ->  ( B  X.  { ~P U. ran  A } )  ~~  B )
2720, 25, 26syl2anc 643 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  ~~  B )
2819, 27jca 519 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319   (/)c0 3628   ~Pcpw 3799   {csn 3814   <.cop 3817   U.cuni 4015   class class class wbr 4212    X. cxp 4876   ran crn 4879   ` cfv 5454   1stc1st 6347   2ndc2nd 6348    ~~ cen 7106
This theorem is referenced by:  disjenex  7265  domss2  7266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1st 6349  df-2nd 6350  df-en 7110
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