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Theorem fbasne0 20059
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 5883 . . . 4  |-  ( F  e.  ( fBas `  B
)  ->  B  e.  dom  fBas )
2 isfbas 20058 . . . 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 16 . . 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 241 . 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 997 . 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 16 1  |-  ( F  e.  ( fBas `  B
)  ->  F  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    e. wcel 1762    =/= wne 2655    e/ wnel 2656   A.wral 2807    i^i cin 3468    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   dom cdm 4992   ` cfv 5579   fBascfbas 18170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-fbas 18180
This theorem is referenced by:  fbdmn0  20063  fbssint  20067  fbun  20069  trfbas2  20072  filtop  20084  fsubbas  20096  fgcl  20107  fbasrn  20113  fmfnfm  20187
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