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Theorem fbasne0 20625
Description: There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
fbasne0  |-  ( F  e.  ( fBas `  B
)  ->  F  =/=  (/) )

Proof of Theorem fbasne0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5877 . . . 4  |-  ( F  e.  ( fBas `  B
)  ->  B  e.  dom  fBas )
2 isfbas 20624 . . . 4  |-  ( B  e.  dom  fBas  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_ 
~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
31, 2syl 17 . . 3  |-  ( F  e.  ( fBas `  B
)  ->  ( F  e.  ( fBas `  B
)  <->  ( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
43ibi 243 . 2  |-  ( F  e.  ( fBas `  B
)  ->  ( F  C_ 
~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) )
5 simpr1 1005 . 2  |-  ( ( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) )  ->  F  =/=  (/) )
64, 5syl 17 1  |-  ( F  e.  ( fBas `  B
)  ->  F  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    e. wcel 1844    =/= wne 2600    e/ wnel 2601   A.wral 2756    i^i cin 3415    C_ wss 3416   (/)c0 3740   ~Pcpw 3957   dom cdm 4825   ` cfv 5571   fBascfbas 18728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fv 5579  df-fbas 18738
This theorem is referenced by:  fbdmn0  20629  fbssint  20633  fbun  20635  trfbas2  20638  filtop  20650  fsubbas  20662  fgcl  20673  fbasrn  20679  fmfnfm  20753
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