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Theorem filss 17548
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 17542 . . . 4  |-  ( F  e.  ( Fil `  X
)  <->  ( F  e.  ( fBas `  X
)  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) ) )
21simprbi 450 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) )
32adantr 451 . 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 5554 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
5 simp2 956 . . 3  |-  ( ( A  e.  F  /\  B  C_  X  /\  A  C_  B )  ->  B  C_  X )
6 elpw2g 4174 . . . 4  |-  ( X  e.  dom  Fil  ->  ( B  e.  ~P X  <->  B 
C_  X ) )
76biimpar 471 . . 3  |-  ( ( X  e.  dom  Fil  /\  B  C_  X )  ->  B  e.  ~P X
)
84, 5, 7syl2an 463 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  B  e.  ~P X
)
9 simpr1 961 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  F )
10 simpr3 963 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  C_  B )
11 elpwg 3632 . . . . 5  |-  ( A  e.  F  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
129, 11syl 15 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  -> 
( A  e.  ~P B 
<->  A  C_  B )
)
1310, 12mpbird 223 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  ~P B
)
14 inelcm 3509 . . 3  |-  ( ( A  e.  F  /\  A  e.  ~P B
)  ->  ( F  i^i  ~P B )  =/=  (/) )
159, 13, 14syl2anc 642 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  -> 
( F  i^i  ~P B )  =/=  (/) )
16 pweq 3628 . . . . . 6  |-  ( x  =  B  ->  ~P x  =  ~P B
)
1716ineq2d 3370 . . . . 5  |-  ( x  =  B  ->  ( F  i^i  ~P x )  =  ( F  i^i  ~P B ) )
1817neeq1d 2459 . . . 4  |-  ( x  =  B  ->  (
( F  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P B )  =/=  (/) ) )
19 eleq1 2343 . . . 4  |-  ( x  =  B  ->  (
x  e.  F  <->  B  e.  F ) )
2018, 19imbi12d 311 . . 3  |-  ( x  =  B  ->  (
( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F )  <->  ( ( F  i^i  ~P B )  =/=  (/)  ->  B  e.  F ) ) )
2120rspccv 2881 . 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 57 1  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  B  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   dom cdm 4689   ` cfv 5255   fBascfbas 17518   Filcfil 17540
This theorem is referenced by:  filin  17549  filtop  17550  isfil2  17551  infil  17558  fgfil  17570  fgabs  17574  filcon  17578  filuni  17580  trfil2  17582  trfg  17586  isufil2  17603  ufprim  17604  ufileu  17614  filufint  17615  elfm3  17645  rnelfm  17648  fmfnfmlem2  17650  fmfnfmlem4  17652  flimopn  17670  flimrest  17678  flimfnfcls  17723  fclscmpi  17724  alexsublem  17738  cfil3i  18695  cfilfcls  18700  iscmet3lem2  18718  equivcfil  18725  relcmpcmet  18742  minveclem4  18796  flfnei2  25577  fgmin  26319
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-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-fil 17541
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