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Theorem ufilmax 17939
Description: Any filter finer than an ultrafilter is actually equal to it. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
ufilmax  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  F  =  G )

Proof of Theorem ufilmax
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3 959 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  F  C_  G
)
2 filelss 17884 . . . . . 6  |-  ( ( G  e.  ( Fil `  X )  /\  x  e.  G )  ->  x  C_  X )
32ex 424 . . . . 5  |-  ( G  e.  ( Fil `  X
)  ->  ( x  e.  G  ->  x  C_  X ) )
433ad2ant2 979 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  x  C_  X ) )
5 ufilb 17938 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  C_  X )  ->  ( -.  x  e.  F  <->  ( X  \  x )  e.  F ) )
653ad2antl1 1119 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( -.  x  e.  F  <->  ( X  \  x )  e.  F
) )
7 simpl3 962 . . . . . . . . . 10  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  ->  F  C_  G )
87sseld 3347 . . . . . . . . 9  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( ( X  \  x )  e.  F  ->  ( X  \  x
)  e.  G ) )
9 filfbas 17880 . . . . . . . . . . . . 13  |-  ( G  e.  ( Fil `  X
)  ->  G  e.  ( fBas `  X )
)
10 fbncp 17871 . . . . . . . . . . . . . 14  |-  ( ( G  e.  ( fBas `  X )  /\  x  e.  G )  ->  -.  ( X  \  x
)  e.  G )
1110ex 424 . . . . . . . . . . . . 13  |-  ( G  e.  ( fBas `  X
)  ->  ( x  e.  G  ->  -.  ( X  \  x )  e.  G ) )
129, 11syl 16 . . . . . . . . . . . 12  |-  ( G  e.  ( Fil `  X
)  ->  ( x  e.  G  ->  -.  ( X  \  x )  e.  G ) )
1312con2d 109 . . . . . . . . . . 11  |-  ( G  e.  ( Fil `  X
)  ->  ( ( X  \  x )  e.  G  ->  -.  x  e.  G ) )
14133ad2ant2 979 . . . . . . . . . 10  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( ( X  \  x )  e.  G  ->  -.  x  e.  G ) )
1514adantr 452 . . . . . . . . 9  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( ( X  \  x )  e.  G  ->  -.  x  e.  G
) )
168, 15syld 42 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( ( X  \  x )  e.  F  ->  -.  x  e.  G
) )
176, 16sylbid 207 . . . . . . 7  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( -.  x  e.  F  ->  -.  x  e.  G ) )
1817con4d 99 . . . . . 6  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( x  e.  G  ->  x  e.  F ) )
1918ex 424 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  C_  X  ->  ( x  e.  G  ->  x  e.  F ) ) )
2019com23 74 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  ( x 
C_  X  ->  x  e.  F ) ) )
214, 20mpdd 38 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  x  e.  F ) )
2221ssrdv 3354 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  G  C_  F
)
231, 22eqssd 3365 1  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  F  =  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3317    C_ wss 3320   ` cfv 5454   fBascfbas 16689   Filcfil 17877   UFilcufil 17931
This theorem is referenced by:  isufil2  17940  ufileu  17951  uffixfr  17955  fmufil  17991  uffclsflim  18063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-fbas 16699  df-fil 17878  df-ufil 17933
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