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Theorem ufilmax 17602
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 957 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  F  C_  G
)
2 filelss 17547 . . . . . 6  |-  ( ( G  e.  ( Fil `  X )  /\  x  e.  G )  ->  x  C_  X )
32ex 423 . . . . 5  |-  ( G  e.  ( Fil `  X
)  ->  ( x  e.  G  ->  x  C_  X ) )
433ad2ant2 977 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  x  C_  X ) )
5 ufilb 17601 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  C_  X )  ->  ( -.  x  e.  F  <->  ( X  \  x )  e.  F ) )
653ad2antl1 1117 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( -.  x  e.  F  <->  ( X  \  x )  e.  F
) )
7 simpl3 960 . . . . . . . . . 10  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  ->  F  C_  G )
87sseld 3179 . . . . . . . . 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 17543 . . . . . . . . . . . . 13  |-  ( G  e.  ( Fil `  X
)  ->  G  e.  ( fBas `  X )
)
10 fbncp 17534 . . . . . . . . . . . . . 14  |-  ( ( G  e.  ( fBas `  X )  /\  x  e.  G )  ->  -.  ( X  \  x
)  e.  G )
1110ex 423 . . . . . . . . . . . . 13  |-  ( G  e.  ( fBas `  X
)  ->  ( x  e.  G  ->  -.  ( X  \  x )  e.  G ) )
129, 11syl 15 . . . . . . . . . . . 12  |-  ( G  e.  ( Fil `  X
)  ->  ( x  e.  G  ->  -.  ( X  \  x )  e.  G ) )
1312con2d 107 . . . . . . . . . . 11  |-  ( G  e.  ( Fil `  X
)  ->  ( ( X  \  x )  e.  G  ->  -.  x  e.  G ) )
14133ad2ant2 977 . . . . . . . . . 10  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( ( X  \  x )  e.  G  ->  -.  x  e.  G ) )
1514adantr 451 . . . . . . . . 9  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( ( X  \  x )  e.  G  ->  -.  x  e.  G
) )
168, 15syld 40 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( ( X  \  x )  e.  F  ->  -.  x  e.  G
) )
176, 16sylbid 206 . . . . . . 7  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( -.  x  e.  F  ->  -.  x  e.  G ) )
1817con4d 97 . . . . . 6  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( x  e.  G  ->  x  e.  F ) )
1918ex 423 . . . . 5  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  C_  X  ->  ( x  e.  G  ->  x  e.  F ) ) )
2019com23 72 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  ( x 
C_  X  ->  x  e.  F ) ) )
214, 20mpdd 36 . . 3  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  x  e.  F ) )
2221ssrdv 3185 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  G  C_  F
)
231, 22eqssd 3196 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   ` cfv 5255   fBascfbas 17518   Filcfil 17540   UFilcufil 17594
This theorem is referenced by:  isufil2  17603  ufileu  17614  uffixfr  17618  fmufil  17654  uffclsflim  17726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-fbas 17520  df-fil 17541  df-ufil 17596
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