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Theorem ufilmax 20143
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 998 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  F  C_  G
)
2 filelss 20088 . . . . . 6  |-  ( ( G  e.  ( Fil `  X )  /\  x  e.  G )  ->  x  C_  X )
32ex 434 . . . . 5  |-  ( G  e.  ( Fil `  X
)  ->  ( x  e.  G  ->  x  C_  X ) )
433ad2ant2 1018 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  x  C_  X ) )
5 ufilb 20142 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  C_  X )  ->  ( -.  x  e.  F  <->  ( X  \  x )  e.  F ) )
653ad2antl1 1158 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( -.  x  e.  F  <->  ( X  \  x )  e.  F
) )
7 simpl3 1001 . . . . . . . . . 10  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  ->  F  C_  G )
87sseld 3503 . . . . . . . . 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 20084 . . . . . . . . . . . . 13  |-  ( G  e.  ( Fil `  X
)  ->  G  e.  ( fBas `  X )
)
10 fbncp 20075 . . . . . . . . . . . . . 14  |-  ( ( G  e.  ( fBas `  X )  /\  x  e.  G )  ->  -.  ( X  \  x
)  e.  G )
1110ex 434 . . . . . . . . . . . . 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 1018 . . . . . . . . . 10  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( ( X  \  x )  e.  G  ->  -.  x  e.  G ) )
1514adantr 465 . . . . . . . . 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 434 . . . . 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 3510 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  G  C_  F
)
231, 22eqssd 3521 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 369    /\ w3a 973    = wceq 1379    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:  isufil2  20144  ufileu  20155  uffixfr  20159  fmufil  20195  uffclsflim  20267
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