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Theorem disjenex 7104
Description: Existence version of disjen 7103. (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 4295 . . 3  |-  { ~P U.
ran  A }  e.  _V
3 xpexg 4879 . . 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 7103 . 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 3440 . . . . 5  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  ( A  i^i  x )  =  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
76eqeq1d 2366 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
( A  i^i  x
)  =  (/)  <->  ( A  i^i  ( B  X.  { ~P U. ran  A }
) )  =  (/) ) )
8 breq1 4105 . . . 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 2945 . 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 1541    = wceq 1642    e. wcel 1710   _Vcvv 2864    i^i cin 3227   (/)c0 3531   ~Pcpw 3701   {csn 3716   U.cuni 3906   class class class wbr 4102    X. cxp 4766   ran crn 4769    ~~ cen 6945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-int 3942  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-1st 6206  df-2nd 6207  df-en 6949
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