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

Theorem ufilss 17938
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 5758 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  X  e.  dom  UFil )
2 elpw2g 4364 . . . 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 17936 . . . . 5  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
54simprbi 452 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) )
6 eleq1 2497 . . . . . 6  |-  ( x  =  S  ->  (
x  e.  F  <->  S  e.  F ) )
7 difeq2 3460 . . . . . . 7  |-  ( x  =  S  ->  ( X  \  x )  =  ( X  \  S
) )
87eleq1d 2503 . . . . . 6  |-  ( x  =  S  ->  (
( X  \  x
)  e.  F  <->  ( X  \  S )  e.  F
) )
96, 8orbi12d 692 . . . . 5  |-  ( x  =  S  ->  (
( x  e.  F  \/  ( X  \  x
)  e.  F )  <-> 
( S  e.  F  \/  ( X  \  S
)  e.  F ) ) )
109rspccv 3050 . . . 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 228 . 2  |-  ( F  e.  ( UFil `  X
)  ->  ( S  C_  X  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) ) )
1312imp 420 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706    \ cdif 3318    C_ wss 3321   ~Pcpw 3800   dom cdm 4879   ` cfv 5455   Filcfil 17878   UFilcufil 17932
This theorem is referenced by:  ufilb  17939  trufil  17943  ufildr  17964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fv 5463  df-ufil 17934
  Copyright terms: Public domain W3C validator