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Theorem 0nelfil 20080
Description: The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
0nelfil  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)

Proof of Theorem 0nelfil
StepHypRef Expression
1 filfbas 20079 . 2  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 0nelfb 20062 . 2  |-  ( F  e.  ( fBas `  X
)  ->  -.  (/)  e.  F
)
31, 2syl 16 1  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1762   (/)c0 3780   ` cfv 5581   fBascfbas 18172   Filcfil 20076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fv 5589  df-fbas 18182  df-fil 20077
This theorem is referenced by:  fileln0  20081  isfil2  20087  infil  20094  filuni  20116  filufint  20151  rnelfmlem  20183  fmfnfm  20189  fclscmpi  20260
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