MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ufilmax Structured version   Unicode version

Theorem ufilmax 19607
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 990 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  F  C_  G
)
2 filelss 19552 . . . . . 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 1010 . . . 4  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  ( x  e.  G  ->  x  C_  X ) )
5 ufilb 19606 . . . . . . . . 9  |-  ( ( F  e.  ( UFil `  X )  /\  x  C_  X )  ->  ( -.  x  e.  F  <->  ( X  \  x )  e.  F ) )
653ad2antl1 1150 . . . . . . . 8  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  -> 
( -.  x  e.  F  <->  ( X  \  x )  e.  F
) )
7 simpl3 993 . . . . . . . . . 10  |-  ( ( ( F  e.  (
UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  /\  x  C_  X )  ->  F  C_  G )
87sseld 3458 . . . . . . . . 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 19548 . . . . . . . . . . . . 13  |-  ( G  e.  ( Fil `  X
)  ->  G  e.  ( fBas `  X )
)
10 fbncp 19539 . . . . . . . . . . . . . 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 1010 . . . . . . . . . 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 3465 . 2  |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X
)  /\  F  C_  G
)  ->  G  C_  F
)
231, 22eqssd 3476 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 965    = wceq 1370    e. wcel 1758    \ cdif 3428    C_ wss 3431   ` cfv 5521   fBascfbas 17924   Filcfil 19545   UFilcufil 19599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529  df-fbas 17934  df-fil 19546  df-ufil 19601
This theorem is referenced by:  isufil2  19608  ufileu  19619  uffixfr  19623  fmufil  19659  uffclsflim  19731
  Copyright terms: Public domain W3C validator