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Theorem disjenex 7019
Description: Existence version of disjen 7018. (Contributed by Mario Carneiro, 7-Feb-2015.)
Assertion
Ref Expression
disjenex  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x ( ( A  i^i  x )  =  (/)  /\  x  ~~  B ) )
Distinct variable groups:    x, A    x, B    x, V    x, W

Proof of Theorem disjenex
StepHypRef Expression
1 simpr 447 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
2 snex 4216 . . 3  |-  { ~P U.
ran  A }  e.  _V
3 xpexg 4800 . . 3  |-  ( ( B  e.  W  /\  { ~P U. ran  A }  e.  _V )  ->  ( B  X.  { ~P U. ran  A }
)  e.  _V )
41, 2, 3sylancl 643 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  e.  _V )
5 disjen 7018 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  i^i  ( B  X.  { ~P U.
ran  A } ) )  =  (/)  /\  ( B  X.  { ~P U. ran  A } )  ~~  B ) )
6 ineq2 3364 . . . . 5  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  ( A  i^i  x )  =  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
76eqeq1d 2291 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
( A  i^i  x
)  =  (/)  <->  ( A  i^i  ( B  X.  { ~P U. ran  A }
) )  =  (/) ) )
8 breq1 4026 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
x  ~~  B  <->  ( B  X.  { ~P U. ran  A } )  ~~  B
) )
97, 8anbi12d 691 . . 3  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
( ( A  i^i  x )  =  (/)  /\  x  ~~  B )  <-> 
( ( A  i^i  ( B  X.  { ~P U.
ran  A } ) )  =  (/)  /\  ( B  X.  { ~P U. ran  A } )  ~~  B ) ) )
109spcegv 2869 . 2  |-  ( ( B  X.  { ~P U.
ran  A } )  e.  _V  ->  (
( ( A  i^i  ( B  X.  { ~P U.
ran  A } ) )  =  (/)  /\  ( B  X.  { ~P U. ran  A } )  ~~  B )  ->  E. x
( ( A  i^i  x )  =  (/)  /\  x  ~~  B ) ) )
114, 5, 10sylc 56 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x ( ( A  i^i  x )  =  (/)  /\  x  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023    X. cxp 4687   ran crn 4690    ~~ cen 6860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-2nd 6123  df-en 6864
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