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Theorem ufilb 20142
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 20141 . . 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 20140 . . . 4  |-  ( F  e.  ( UFil `  X
)  ->  F  e.  ( Fil `  X ) )
4 filfbas 20084 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
5 fbncp 20075 . . . . . 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 1767    \ cdif 3473    C_ wss 3476   ` cfv 5586   fBascfbas 18177   Filcfil 20081   UFilcufil 20135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-fbas 18187  df-fil 20082  df-ufil 20137
This theorem is referenced by:  ufilmax  20143  ufprim  20145  trufil  20146  ufileu  20155  cfinufil  20164  alexsublem  20279
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