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Theorem ufilb 20533
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 20532 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F ) )
21ord 377 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F  ->  ( X  \  S
)  e.  F ) )
3 ufilfil 20531 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
4 filfbas 20475 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
5 fbncp 20466 . . . . . 6  |-  ( ( F  e.  ( fBas `  X )  /\  S  e.  F )  ->  -.  ( X  \  S )  e.  F )
65ex 434 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  ( S  e.  F  ->  -.  ( X  \  S )  e.  F ) )
76con2d 115 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( ( X  \  S )  e.  F  ->  -.  S  e.  F ) )
83, 4, 73syl 20 . . 3  |-  ( F  e.  ( UFil `  X
)  ->  ( ( X  \  S )  e.  F  ->  -.  S  e.  F ) )
98adantr 465 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  (
( X  \  S
)  e.  F  ->  -.  S  e.  F
) )
102, 9impbid 191 1  |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F  <->  ( X  \  S )  e.  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1819    \ cdif 3468    C_ wss 3471   ` cfv 5594   fBascfbas 18533   Filcfil 20472   UFilcufil 20526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-fbas 18543  df-fil 20473  df-ufil 20528
This theorem is referenced by:  ufilmax  20534  ufprim  20536  trufil  20537  ufileu  20546  cfinufil  20555  alexsublem  20670
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