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Theorem filn0 19577
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 19570 . 2  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
2 ne0i 3754 . 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 1758    =/= wne 2648   (/)c0 3748   ` cfv 5529   Filcfil 19560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-fbas 17949  df-fil 19561
This theorem is referenced by:  ufileu  19634  filufint  19635  uffixfr  19638  uffix2  19639  uffixsn  19640  hausflim  19696  fclsval  19723  isfcls  19724  fclsopn  19729  fclsfnflim  19742  filnetlem4  28773
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