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Theorem filn0 17855
Description: The empty set is not a filter. Remark below def. 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filn0  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )

Proof of Theorem filn0
StepHypRef Expression
1 filtop 17848 . 2  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
2 ne0i 3602 . 2  |-  ( X  e.  F  ->  F  =/=  (/) )
31, 2syl 16 1  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    =/= wne 2575   (/)c0 3596   ` cfv 5421   Filcfil 17838
This theorem is referenced by:  ufileu  17912  filufint  17913  uffixfr  17916  uffix2  17917  uffixsn  17918  hausflim  17974  fclsval  18001  isfcls  18002  fclsopn  18007  fclsfnflim  18020  filnetlem4  26308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fv 5429  df-fbas 16662  df-fil 17839
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