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Theorem filss 20332
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 20326 . . . 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 5882 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  dom  Fil )
5 simp2 998 . . 3  |-  ( ( A  e.  F  /\  B  C_  X  /\  A  C_  B )  ->  B  C_  X )
6 elpw2g 4600 . . . 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 1003 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  e.  F )
10 simpr3 1005 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) )  ->  A  C_  B )
11 elpwg 4005 . . . . 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 3867 . . 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 4000 . . . . . 6  |-  ( x  =  B  ->  ~P x  =  ~P B
)
1716ineq2d 3685 . . . . 5  |-  ( x  =  B  ->  ( F  i^i  ~P x )  =  ( F  i^i  ~P B ) )
1817neeq1d 2720 . . . 4  |-  ( x  =  B  ->  (
( F  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P B )  =/=  (/) ) )
19 eleq1 2515 . . . 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 3193 . 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793    i^i cin 3460    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   dom cdm 4989   ` cfv 5578   fBascfbas 18385   Filcfil 20324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fv 5586  df-fil 20325
This theorem is referenced by:  filin  20333  filtop  20334  isfil2  20335  infil  20342  fgfil  20354  fgabs  20358  filcon  20362  filuni  20364  trfil2  20366  trfg  20370  isufil2  20387  ufprim  20388  ufileu  20398  filufint  20399  elfm3  20429  rnelfm  20432  fmfnfmlem2  20434  fmfnfmlem4  20436  flimopn  20454  flimrest  20462  flimfnfcls  20507  fclscmpi  20508  alexsublem  20522  metustOLD  21048  metust  21049  cfil3i  21686  cfilfcls  21691  iscmet3lem2  21709  equivcfil  21716  relcmpcmet  21733  minveclem4  21825  fgmin  30164
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