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Theorem ufilmax 20493
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 996 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  F  C_  G
)
2 filelss 20438 . . . . . 6  |-  ( ( G  e.  ( Fil `  X )  /\  x  e.  G )  ->  x  C_  X )
32ex 432 . . . . 5  |-  ( G  e.  ( Fil `  X
)  ->  ( x  e.  G  ->  x  C_  X ) )
433ad2ant2 1016 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  x  C_  X ) )
5 ufilb 20492 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  C_  X )  ->  ( -.  x  e.  F  <->  ( X  \  x )  e.  F ) )
653ad2antl1 1156 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( -.  x  e.  F  <->  ( X  \  x )  e.  F
) )
7 simpl3 999 . . . . . . . . . 10  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  ->  F  C_  G )
87sseld 3416 . . . . . . . . 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 20434 . . . . . . . . . . . . 13  |-  ( G  e.  ( Fil `  X
)  ->  G  e.  ( fBas `  X )
)
10 fbncp 20425 . . . . . . . . . . . . . 14  |-  ( ( G  e.  ( fBas `  X )  /\  x  e.  G )  ->  -.  ( X  \  x
)  e.  G )
1110ex 432 . . . . . . . . . . . . 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 115 . . . . . . . . . . 11  |-  ( G  e.  ( Fil `  X
)  ->  ( ( X  \  x )  e.  G  ->  -.  x  e.  G ) )
14133ad2ant2 1016 . . . . . . . . . 10  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( ( X  \  x )  e.  G  ->  -.  x  e.  G ) )
1514adantr 463 . . . . . . . . 9  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( ( X  \  x )  e.  G  ->  -.  x  e.  G
) )
168, 15syld 44 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( ( X  \  x )  e.  F  ->  -.  x  e.  G
) )
176, 16sylbid 215 . . . . . . 7  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( -.  x  e.  F  ->  -.  x  e.  G ) )
1817con4d 105 . . . . . 6  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( x  e.  G  ->  x  e.  F ) )
1918ex 432 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  C_  X  ->  ( x  e.  G  ->  x  e.  F ) ) )
2019com23 78 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  ( x 
C_  X  ->  x  e.  F ) ) )
214, 20mpdd 40 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  x  e.  F ) )
2221ssrdv 3423 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  G  C_  F
)
231, 22eqssd 3434 1  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    \ cdif 3386    C_ wss 3389   ` cfv 5496   fBascfbas 18519   Filcfil 20431   UFilcufil 20485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fv 5504  df-fbas 18529  df-fil 20432  df-ufil 20487
This theorem is referenced by:  isufil2  20494  ufileu  20505  uffixfr  20509  fmufil  20545  uffclsflim  20617
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