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Theorem ufilss 17616
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 5570 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  X  e.  dom  UFil )
2 elpw2g 4190 . . . 4  |-  ( X  e.  dom  UFil  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
31, 2syl 15 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( S  e.  ~P X  <->  S  C_  X
) )
4 isufil 17614 . . . . 5  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
54simprbi 450 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) )
6 eleq1 2356 . . . . . 6  |-  ( x  =  S  ->  (
x  e.  F  <->  S  e.  F ) )
7 difeq2 3301 . . . . . . 7  |-  ( x  =  S  ->  ( X  \  x )  =  ( X  \  S
) )
87eleq1d 2362 . . . . . 6  |-  ( x  =  S  ->  (
( X  \  x
)  e.  F  <->  ( X  \  S )  e.  F
) )
96, 8orbi12d 690 . . . . 5  |-  ( x  =  S  ->  (
( x  e.  F  \/  ( X  \  x
)  e.  F )  <-> 
( S  e.  F  \/  ( X  \  S
)  e.  F ) ) )
109rspccv 2894 . . . 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 15 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( S  e.  ~P X  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) ) )
123, 11sylbird 226 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( S  C_  X  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) ) )
1312imp 418 1  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    C_ wss 3165   ~Pcpw 3638   dom cdm 4705   ` cfv 5271   Filcfil 17556   UFilcufil 17610
This theorem is referenced by:  ufilb  17617  trufil  17621  ufildr  17642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ufil 17612
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