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Theorem fbdmn0 17787
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 17784 . 2  |-  ( F  e.  ( fBas `  B
)  ->  -.  (/)  e.  F
)
2 fveq2 5668 . . . . . 6  |-  ( B  =  (/)  ->  ( fBas `  B )  =  (
fBas `  (/) ) )
32eleq2d 2454 . . . . 5  |-  ( B  =  (/)  ->  ( F  e.  ( fBas `  B
)  <->  F  e.  ( fBas `  (/) ) ) )
43biimpd 199 . . . 4  |-  ( B  =  (/)  ->  ( F  e.  ( fBas `  B
)  ->  F  e.  ( fBas `  (/) ) ) )
5 fbasne0 17783 . . . . . 6  |-  ( F  e.  ( fBas `  (/) )  ->  F  =/=  (/) )
6 n0 3580 . . . . . 6  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
75, 6sylib 189 . . . . 5  |-  ( F  e.  ( fBas `  (/) )  ->  E. x  x  e.  F )
8 fbelss 17786 . . . . . . 7  |-  ( ( F  e.  ( fBas `  (/) )  /\  x  e.  F )  ->  x  C_  (/) )
9 ss0 3601 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  =  (/) )
108, 9syl 16 . . . . . 6  |-  ( ( F  e.  ( fBas `  (/) )  /\  x  e.  F )  ->  x  =  (/) )
11 simpr 448 . . . . . 6  |-  ( ( F  e.  ( fBas `  (/) )  /\  x  e.  F )  ->  x  e.  F )
1210, 11eqeltrrd 2462 . . . . 5  |-  ( ( F  e.  ( fBas `  (/) )  /\  x  e.  F )  ->  (/)  e.  F
)
137, 12exlimddv 1645 . . . 4  |-  ( F  e.  ( fBas `  (/) )  ->  (/) 
e.  F )
144, 13syl6com 33 . . 3  |-  ( F  e.  ( fBas `  B
)  ->  ( B  =  (/)  ->  (/)  e.  F
) )
1514necon3bd 2587 . 2  |-  ( F  e.  ( fBas `  B
)  ->  ( -.  (/) 
e.  F  ->  B  =/=  (/) ) )
161, 15mpd 15 1  |-  ( F  e.  ( fBas `  B
)  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2550    C_ wss 3263   (/)c0 3571   ` cfv 5394   fBascfbas 16615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402  df-fbas 16623
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