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Theorem fileln0 20520
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 20519 . 2  |-  ( F  e.  ( Fil `  X
)  ->  -.  (/)  e.  F
)
3 nelne2 2784 . 2  |-  ( ( A  e.  F  /\  -.  (/)  e.  F )  ->  A  =/=  (/) )
41, 2, 3syl2anr 476 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    e. wcel 1823    =/= wne 2649   (/)c0 3783   ` cfv 5570   Filcfil 20515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-fbas 18614  df-fil 20516
This theorem is referenced by:  filinn0  20530  filintn0  20531  alexsublem  20713  cfil3i  21877  iscmet3  21901  filnetlem4  30442
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