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Theorem fbdmn0 17529
Description: The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
fbdmn0  |-  ( F  e.  ( fBas `  B
)  ->  B  =/=  (/) )

Proof of Theorem fbdmn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0nelfb 17526 . 2  |-  ( F  e.  ( fBas `  B
)  ->  -.  (/)  e.  F
)
2 fveq2 5525 . . . . . 6  |-  ( B  =  (/)  ->  ( fBas `  B )  =  (
fBas `  (/) ) )
32eleq2d 2350 . . . . 5  |-  ( B  =  (/)  ->  ( F  e.  ( fBas `  B
)  <->  F  e.  ( fBas `  (/) ) ) )
43biimpd 198 . . . 4  |-  ( B  =  (/)  ->  ( F  e.  ( fBas `  B
)  ->  F  e.  ( fBas `  (/) ) ) )
5 fbasne0 17525 . . . . . 6  |-  ( F  e.  ( fBas `  (/) )  ->  F  =/=  (/) )
6 n0 3464 . . . . . 6  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
75, 6sylib 188 . . . . 5  |-  ( F  e.  ( fBas `  (/) )  ->  E. x  x  e.  F )
8 fbelss 17528 . . . . . . . . 9  |-  ( ( F  e.  ( fBas `  (/) )  /\  x  e.  F )  ->  x  C_  (/) )
9 ss0 3485 . . . . . . . . 9  |-  ( x 
C_  (/)  ->  x  =  (/) )
108, 9syl 15 . . . . . . . 8  |-  ( ( F  e.  ( fBas `  (/) )  /\  x  e.  F )  ->  x  =  (/) )
11 simpr 447 . . . . . . . 8  |-  ( ( F  e.  ( fBas `  (/) )  /\  x  e.  F )  ->  x  e.  F )
1210, 11eqeltrrd 2358 . . . . . . 7  |-  ( ( F  e.  ( fBas `  (/) )  /\  x  e.  F )  ->  (/)  e.  F
)
1312ex 423 . . . . . 6  |-  ( F  e.  ( fBas `  (/) )  -> 
( x  e.  F  -> 
(/)  e.  F )
)
1413exlimdv 1664 . . . . 5  |-  ( F  e.  ( fBas `  (/) )  -> 
( E. x  x  e.  F  ->  (/)  e.  F
) )
157, 14mpd 14 . . . 4  |-  ( F  e.  ( fBas `  (/) )  ->  (/) 
e.  F )
164, 15syl6com 31 . . 3  |-  ( F  e.  ( fBas `  B
)  ->  ( B  =  (/)  ->  (/)  e.  F
) )
1716necon3bd 2483 . 2  |-  ( F  e.  ( fBas `  B
)  ->  ( -.  (/) 
e.  F  ->  B  =/=  (/) ) )
181, 17mpd 14 1  |-  ( F  e.  ( fBas `  B
)  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   ` cfv 5255   fBascfbas 17518
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
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