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Theorem filss 19426
Description: A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filss  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  B  e.  F )

Proof of Theorem filss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfil 19420 . . . 4  |-  ( F  e.  ( Fil `  X
)  <->  ( F  e.  ( fBas `  X
)  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) ) )
21simprbi 464 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) )
32adantr 465 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A. x  e.  ~P  X ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) )
4 elfvdm 5716 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
5 simp2 989 . . 3  |-  ( ( A  e.  F  /\  B  C_  X  /\  A  C_  B )  ->  B  C_  X )
6 elpw2g 4455 . . . 4  |-  ( X  e.  dom  Fil  ->  ( B  e.  ~P X  <->  B 
C_  X ) )
76biimpar 485 . . 3  |-  ( ( X  e.  dom  Fil  /\  B  C_  X )  ->  B  e.  ~P X
)
84, 5, 7syl2an 477 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  B  e.  ~P X
)
9 simpr1 994 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  F )
10 simpr3 996 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  C_  B )
11 elpwg 3868 . . . . 5  |-  ( A  e.  F  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
129, 11syl 16 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  -> 
( A  e.  ~P B 
<->  A  C_  B )
)
1310, 12mpbird 232 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  ~P B
)
14 inelcm 3733 . . 3  |-  ( ( A  e.  F  /\  A  e.  ~P B
)  ->  ( F  i^i  ~P B )  =/=  (/) )
159, 13, 14syl2anc 661 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  -> 
( F  i^i  ~P B )  =/=  (/) )
16 pweq 3863 . . . . . 6  |-  ( x  =  B  ->  ~P x  =  ~P B
)
1716ineq2d 3552 . . . . 5  |-  ( x  =  B  ->  ( F  i^i  ~P x )  =  ( F  i^i  ~P B ) )
1817neeq1d 2621 . . . 4  |-  ( x  =  B  ->  (
( F  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P B )  =/=  (/) ) )
19 eleq1 2503 . . . 4  |-  ( x  =  B  ->  (
x  e.  F  <->  B  e.  F ) )
2018, 19imbi12d 320 . . 3  |-  ( x  =  B  ->  (
( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F )  <->  ( ( F  i^i  ~P B )  =/=  (/)  ->  B  e.  F ) ) )
2120rspccv 3070 . 2  |-  ( A. x  e.  ~P  X
( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F )  ->  ( B  e.  ~P X  ->  ( ( F  i^i  ~P B )  =/=  (/)  ->  B  e.  F ) ) )
223, 8, 15, 21syl3c 61 1  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  B  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   dom cdm 4840   ` cfv 5418   fBascfbas 17804   Filcfil 19418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fv 5426  df-fil 19419
This theorem is referenced by:  filin  19427  filtop  19428  isfil2  19429  infil  19436  fgfil  19448  fgabs  19452  filcon  19456  filuni  19458  trfil2  19460  trfg  19464  isufil2  19481  ufprim  19482  ufileu  19492  filufint  19493  elfm3  19523  rnelfm  19526  fmfnfmlem2  19528  fmfnfmlem4  19530  flimopn  19548  flimrest  19556  flimfnfcls  19601  fclscmpi  19602  alexsublem  19616  metustOLD  20142  metust  20143  cfil3i  20780  cfilfcls  20785  iscmet3lem2  20803  equivcfil  20810  relcmpcmet  20827  minveclem4  20919  fgmin  28591
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