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Theorem filss 17916
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 17910 . . . 4  |-  ( F  e.  ( Fil `  X
)  <->  ( F  e.  ( fBas `  X
)  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) ) )
21simprbi 452 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) )
32adantr 453 . 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 5786 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
5 simp2 959 . . 3  |-  ( ( A  e.  F  /\  B  C_  X  /\  A  C_  B )  ->  B  C_  X )
6 elpw2g 4392 . . . 4  |-  ( X  e.  dom  Fil  ->  ( B  e.  ~P X  <->  B 
C_  X ) )
76biimpar 473 . . 3  |-  ( ( X  e.  dom  Fil  /\  B  C_  X )  ->  B  e.  ~P X
)
84, 5, 7syl2an 465 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  B  e.  ~P X
)
9 simpr1 964 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  F )
10 simpr3 966 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  C_  B )
11 elpwg 3830 . . . . 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 225 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  ~P B
)
14 inelcm 3706 . . 3  |-  ( ( A  e.  F  /\  A  e.  ~P B
)  ->  ( F  i^i  ~P B )  =/=  (/) )
159, 13, 14syl2anc 644 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  -> 
( F  i^i  ~P B )  =/=  (/) )
16 pweq 3826 . . . . . 6  |-  ( x  =  B  ->  ~P x  =  ~P B
)
1716ineq2d 3528 . . . . 5  |-  ( x  =  B  ->  ( F  i^i  ~P x )  =  ( F  i^i  ~P B ) )
1817neeq1d 2620 . . . 4  |-  ( x  =  B  ->  (
( F  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P B )  =/=  (/) ) )
19 eleq1 2502 . . . 4  |-  ( x  =  B  ->  (
x  e.  F  <->  B  e.  F ) )
2018, 19imbi12d 313 . . 3  |-  ( x  =  B  ->  (
( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F )  <->  ( ( F  i^i  ~P B )  =/=  (/)  ->  B  e.  F ) ) )
2120rspccv 3055 . 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 60 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711    i^i cin 3305    C_ wss 3306   (/)c0 3613   ~Pcpw 3823   dom cdm 4907   ` cfv 5483   fBascfbas 16720   Filcfil 17908
This theorem is referenced by:  filin  17917  filtop  17918  isfil2  17919  infil  17926  fgfil  17938  fgabs  17942  filcon  17946  filuni  17948  trfil2  17950  trfg  17954  isufil2  17971  ufprim  17972  ufileu  17982  filufint  17983  elfm3  18013  rnelfm  18016  fmfnfmlem2  18018  fmfnfmlem4  18020  flimopn  18038  flimrest  18046  flimfnfcls  18091  fclscmpi  18092  alexsublem  18106  metustOLD  18628  metust  18629  cfil3i  19253  cfilfcls  19258  iscmet3lem2  19276  equivcfil  19283  relcmpcmet  19300  minveclem4  19364  fgmin  26437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fv 5491  df-fil 17909
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