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Theorem filtop 20334
Description: The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filtop  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )

Proof of Theorem filtop
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 filfbas 20327 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbasne0 20309 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  F  =/=  (/) )
31, 2syl 16 . 2  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
4 n0 3780 . . 3  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
5 filelss 20331 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  x  C_  X )
6 ssid 3508 . . . . . . 7  |-  X  C_  X
7 filss 20332 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  X  C_  X  /\  x  C_  X ) )  ->  X  e.  F )
873exp2 1215 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( X 
C_  X  ->  (
x  C_  X  ->  X  e.  F ) ) ) )
98imp 429 . . . . . . 7  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  ( X  C_  X  ->  (
x  C_  X  ->  X  e.  F ) ) )
106, 9mpi 17 . . . . . 6  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  (
x  C_  X  ->  X  e.  F ) )
115, 10mpd 15 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  x  e.  F )  ->  X  e.  F )
1211ex 434 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  X  e.  F ) )
1312exlimdv 1711 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  x  e.  F  ->  X  e.  F ) )
144, 13syl5bi 217 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( F  =/=  (/)  ->  X  e.  F ) )
153, 14mpd 15 1  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1599    e. wcel 1804    =/= wne 2638    C_ wss 3461   (/)c0 3770   ` 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-nel 2641  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-fbas 18395  df-fil 20325
This theorem is referenced by:  isfil2  20335  filn0  20341  infil  20342  filunibas  20360  filuni  20364  trfil1  20365  trfil2  20366  fgtr  20369  trfg  20370  isufil2  20387  filssufil  20391  ssufl  20397  ufileu  20398  filufint  20399  uffixfr  20402  cfinufil  20407  rnelfmlem  20431  rnelfm  20432  fmfnfmlem1  20433  fmfnfmlem2  20434  fmfnfmlem4  20436  fmfnfm  20437  flfval  20469  fclsfnflim  20506  flimfnfcls  20507  fcfval  20512  alexsublem  20522  metustOLD  21048  metust  21049  cmetss  21731  minveclem4a  21823  filnetlem3  30174  filnetlem4  30175
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