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Theorem filss 20946
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 20940 . . . 4  |-  ( F  e.  ( Fil `  X
)  <->  ( F  e.  ( fBas `  X
)  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) ) )
21simprbi 471 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  A. x  e.  ~P  X ( ( F  i^i  ~P x
)  =/=  (/)  ->  x  e.  F ) )
32adantr 472 . 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 5905 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
5 simp2 1031 . . 3  |-  ( ( A  e.  F  /\  B  C_  X  /\  A  C_  B )  ->  B  C_  X )
6 elpw2g 4564 . . . 4  |-  ( X  e.  dom  Fil  ->  ( B  e.  ~P X  <->  B 
C_  X ) )
76biimpar 493 . . 3  |-  ( ( X  e.  dom  Fil  /\  B  C_  X )  ->  B  e.  ~P X
)
84, 5, 7syl2an 485 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  B  e.  ~P X
)
9 simpr1 1036 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  F )
10 simpr3 1038 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  C_  B )
11 elpwg 3950 . . . . 5  |-  ( A  e.  F  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
129, 11syl 17 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  -> 
( A  e.  ~P B 
<->  A  C_  B )
)
1310, 12mpbird 240 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  ~P B
)
14 inelcm 3823 . . 3  |-  ( ( A  e.  F  /\  A  e.  ~P B
)  ->  ( F  i^i  ~P B )  =/=  (/) )
159, 13, 14syl2anc 673 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  -> 
( F  i^i  ~P B )  =/=  (/) )
16 pweq 3945 . . . . . 6  |-  ( x  =  B  ->  ~P x  =  ~P B
)
1716ineq2d 3625 . . . . 5  |-  ( x  =  B  ->  ( F  i^i  ~P x )  =  ( F  i^i  ~P B ) )
1817neeq1d 2702 . . . 4  |-  ( x  =  B  ->  (
( F  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P B )  =/=  (/) ) )
19 eleq1 2537 . . . 4  |-  ( x  =  B  ->  (
x  e.  F  <->  B  e.  F ) )
2018, 19imbi12d 327 . . 3  |-  ( x  =  B  ->  (
( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F )  <->  ( ( F  i^i  ~P B )  =/=  (/)  ->  B  e.  F ) ) )
2120rspccv 3133 . 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 62 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   dom cdm 4839   ` cfv 5589   fBascfbas 19035   Filcfil 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-fil 20939
This theorem is referenced by:  filin  20947  filtop  20948  isfil2  20949  infil  20956  fgfil  20968  fgabs  20972  filcon  20976  filuni  20978  trfil2  20980  trfg  20984  isufil2  21001  ufprim  21002  ufileu  21012  filufint  21013  elfm3  21043  rnelfm  21046  fmfnfmlem2  21048  fmfnfmlem4  21050  flimopn  21068  flimrest  21076  flimfnfcls  21121  fclscmpi  21122  alexsublem  21137  metust  21651  cfil3i  22317  cfilfcls  22322  iscmet3lem2  22340  equivcfil  22347  relcmpcmet  22364  minveclem4  22452  minveclem4OLD  22464  fgmin  31097
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