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Theorem ufilss 19591
Description: For any subset of the base set of an ultrafilter, either the set is in the ultrafilter or the complement is. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
ufilss  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) )

Proof of Theorem ufilss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfvdm 5812 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  X  e.  dom  UFil )
2 elpw2g 4550 . . . 4  |-  ( X  e.  dom  UFil  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
31, 2syl 16 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( S  e.  ~P X  <->  S  C_  X
) )
4 isufil 19589 . . . . 5  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
54simprbi 464 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) )
6 eleq1 2521 . . . . . 6  |-  ( x  =  S  ->  (
x  e.  F  <->  S  e.  F ) )
7 difeq2 3563 . . . . . . 7  |-  ( x  =  S  ->  ( X  \  x )  =  ( X  \  S
) )
87eleq1d 2519 . . . . . 6  |-  ( x  =  S  ->  (
( X  \  x
)  e.  F  <->  ( X  \  S )  e.  F
) )
96, 8orbi12d 709 . . . . 5  |-  ( x  =  S  ->  (
( x  e.  F  \/  ( X  \  x
)  e.  F )  <-> 
( S  e.  F  \/  ( X  \  S
)  e.  F ) ) )
109rspccv 3163 . . . 4  |-  ( A. x  e.  ~P  X
( x  e.  F  \/  ( X  \  x
)  e.  F )  ->  ( S  e. 
~P X  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) ) )
115, 10syl 16 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( S  e.  ~P X  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) ) )
123, 11sylbird 235 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( S  C_  X  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) ) )
1312imp 429 1  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2793    \ cdif 3420    C_ wss 3423   ~Pcpw 3955   dom cdm 4935   ` cfv 5513   Filcfil 19531   UFilcufil 19585
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 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fv 5521  df-ufil 19587
This theorem is referenced by:  ufilb  19592  trufil  19596  ufildr  19617
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