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Theorem filn0 20236
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 20229 . 2  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
2 ne0i 3776 . 2  |-  ( X  e.  F  ->  F  =/=  (/) )
31, 2syl 16 1  |-  ( F  e.  ( Fil `  X
)  ->  F  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1804    =/= wne 2638   (/)c0 3770   ` cfv 5578   Filcfil 20219
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 18290  df-fil 20220
This theorem is referenced by:  ufileu  20293  filufint  20294  uffixfr  20297  uffix2  20298  uffixsn  20299  hausflim  20355  fclsval  20382  isfcls  20383  fclsopn  20388  fclsfnflim  20401  filnetlem4  30174
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