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Theorem disjenex 7582
Description: Existence version of disjen 7581. (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 461 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
2 snex 4644 . . 3  |-  { ~P U.
ran  A }  e.  _V
3 xpexg 6620 . . 3  |-  ( ( B  e.  W  /\  { ~P U. ran  A }  e.  _V )  ->  ( B  X.  { ~P U. ran  A }
)  e.  _V )
41, 2, 3sylancl 662 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  e.  _V )
5 disjen 7581 . 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 3657 . . . . 5  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  ( A  i^i  x )  =  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
76eqeq1d 2456 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
( A  i^i  x
)  =  (/)  <->  ( A  i^i  ( B  X.  { ~P U. ran  A }
) )  =  (/) ) )
8 breq1 4406 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
x  ~~  B  <->  ( B  X.  { ~P U. ran  A } )  ~~  B
) )
97, 8anbi12d 710 . . 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 3164 . 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 60 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x ( ( A  i^i  x )  =  (/)  /\  x  ~~  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3078    i^i cin 3438   (/)c0 3748   ~Pcpw 3971   {csn 3988   U.cuni 4202   class class class wbr 4403    X. cxp 4949   ran crn 4952    ~~ cen 7420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-1st 6690  df-2nd 6691  df-en 7424
This theorem is referenced by: (None)
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