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Theorem filinn0 17555
Description: The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filinn0  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  =/=  (/) )

Proof of Theorem filinn0
StepHypRef Expression
1 simp1 955 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  F  e.  ( Fil `  X
) )
2 filin 17549 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  e.  F )
3 fileln0 17545 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  i^i  B )  e.  F )  ->  ( A  i^i  B )  =/=  (/) )
41, 2, 3syl2anc 642 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684    =/= wne 2446    i^i cin 3151   (/)c0 3455   ` cfv 5255   Filcfil 17540
This theorem is referenced by:  flimclsi  17673  hausflimlem  17674  filnetlem3  26329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-fbas 17520  df-fil 17541
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