MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjenex Structured version   Unicode version

Theorem disjenex 7268
Description: Existence version of disjen 7267. (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 449 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
2 snex 4408 . . 3  |-  { ~P U.
ran  A }  e.  _V
3 xpexg 4992 . . 3  |-  ( ( B  e.  W  /\  { ~P U. ran  A }  e.  _V )  ->  ( B  X.  { ~P U. ran  A }
)  e.  _V )
41, 2, 3sylancl 645 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { ~P U. ran  A }
)  e.  _V )
5 disjen 7267 . 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 3538 . . . . 5  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  ( A  i^i  x )  =  ( A  i^i  ( B  X.  { ~P U. ran  A } ) ) )
76eqeq1d 2446 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
( A  i^i  x
)  =  (/)  <->  ( A  i^i  ( B  X.  { ~P U. ran  A }
) )  =  (/) ) )
8 breq1 4218 . . . 4  |-  ( x  =  ( B  X.  { ~P U. ran  A } )  ->  (
x  ~~  B  <->  ( B  X.  { ~P U. ran  A } )  ~~  B
) )
97, 8anbi12d 693 . . 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 3039 . 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 59 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 360   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017   class class class wbr 4215    X. cxp 4879   ran crn 4882    ~~ cen 7109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-1st 6352  df-2nd 6353  df-en 7113
  Copyright terms: Public domain W3C validator