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Theorem disjenex 7224
Description: Existence version of disjen 7223. (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 448 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
2 snex 4365 . . 3  |-  { ~P U.
ran  A }  e.  _V
3 xpexg 4948 . . 3  |-  ( ( B  e.  W  /\  { ~P U. ran  A }  e.  _V )  ->  ( B  X.  { ~P U. ran  A }
)  e.  _V )
41, 2, 3sylancl 644 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  e.  _V )
5 disjen 7223 . 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 3496 . . . . 5  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  ( A  i^i  x )  =  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
76eqeq1d 2412 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
( A  i^i  x
)  =  (/)  <->  ( A  i^i  ( B  X.  { ~P U. ran  A }
) )  =  (/) ) )
8 breq1 4175 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
x  ~~  B  <->  ( B  X.  { ~P U. ran  A } )  ~~  B
) )
97, 8anbi12d 692 . . 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 2997 . 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 58 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 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279   (/)c0 3588   ~Pcpw 3759   {csn 3774   U.cuni 3975   class class class wbr 4172    X. cxp 4835   ran crn 4838    ~~ cen 7065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1st 6308  df-2nd 6309  df-en 7069
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