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Theorem fileln0 19541
Description: An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
fileln0  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  =/=  (/) )

Proof of Theorem fileln0
StepHypRef Expression
1 id 22 . 2  |-  ( A  e.  F  ->  A  e.  F )
2 0nelfil 19540 . 2  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)
3 nelne2 2778 . 2  |-  ( ( A  e.  F  /\  -.  (/)  e.  F )  ->  A  =/=  (/) )
41, 2, 3syl2anr 478 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1758    =/= wne 2644   (/)c0 3737   ` cfv 5518   Filcfil 19536
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 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fv 5526  df-fbas 17925  df-fil 19537
This theorem is referenced by:  filinn0  19551  filintn0  19552  alexsublem  19734  cfil3i  20898  iscmet3  20922  filnetlem4  28742
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