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Theorem ufilb 17940
Description: The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
ufilb  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F  <->  ( X  \  S )  e.  F ) )

Proof of Theorem ufilb
StepHypRef Expression
1 ufilss 17939 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) )
21ord 368 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F  ->  ( X  \  S
)  e.  F ) )
3 ufilfil 17938 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
4 filfbas 17882 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
5 fbncp 17873 . . . . . 6  |-  ( ( F  e.  ( fBas `  X )  /\  S  e.  F )  ->  -.  ( X  \  S )  e.  F )
65ex 425 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  ( S  e.  F  ->  -.  ( X  \  S )  e.  F ) )
76con2d 110 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( ( X  \  S )  e.  F  ->  -.  S  e.  F ) )
83, 4, 73syl 19 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( ( X  \  S )  e.  F  ->  -.  S  e.  F ) )
98adantr 453 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  (
( X  \  S
)  e.  F  ->  -.  S  e.  F
) )
102, 9impbid 185 1  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F  <->  ( X  \  S )  e.  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726    \ cdif 3319    C_ wss 3322   ` cfv 5456   fBascfbas 16691   Filcfil 17879   UFilcufil 17933
This theorem is referenced by:  ufilmax  17941  ufprim  17943  trufil  17944  ufileu  17953  cfinufil  17962  alexsublem  18077
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 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
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-csb 3254  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-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-fbas 16701  df-fil 17880  df-ufil 17935
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