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Theorem ufilss 20575
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 5874 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  X  e.  dom  UFil )
2 elpw2g 4600 . . . 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 20573 . . . . 5  |-  ( F  e.  ( UFil `  X
)  <->  ( F  e.  ( Fil `  X
)  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) ) )
54simprbi 462 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  A. x  e.  ~P  X ( x  e.  F  \/  ( X  \  x )  e.  F ) )
6 eleq1 2526 . . . . . 6  |-  ( x  =  S  ->  (
x  e.  F  <->  S  e.  F ) )
7 difeq2 3602 . . . . . . 7  |-  ( x  =  S  ->  ( X  \  x )  =  ( X  \  S
) )
87eleq1d 2523 . . . . . 6  |-  ( x  =  S  ->  (
( X  \  x
)  e.  F  <->  ( X  \  S )  e.  F
) )
96, 8orbi12d 707 . . . . 5  |-  ( x  =  S  ->  (
( x  e.  F  \/  ( X  \  x
)  e.  F )  <-> 
( S  e.  F  \/  ( X  \  S
)  e.  F ) ) )
109rspccv 3204 . . . 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 427 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    \ cdif 3458    C_ wss 3461   ~Pcpw 3999   dom cdm 4988   ` cfv 5570   Filcfil 20515   UFilcufil 20569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-ufil 20571
This theorem is referenced by:  ufilb  20576  trufil  20580  ufildr  20601
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