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Theorem 0nelfb 19517
Description: No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
0nelfb  |-  ( F  e.  ( fBas `  B
)  ->  -.  (/)  e.  F
)

Proof of Theorem 0nelfb
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5812 . . . . 5  |-  ( F  e.  ( fBas `  B
)  ->  B  e.  dom  fBas )
2 isfbas 19515 . . . . 5  |-  ( 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 . . . 4  |-  ( 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 . . 3  |-  ( 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 simpr2 995 . . 3  |-  ( ( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) )  ->  (/)  e/  F )
64, 5syl 16 . 2  |-  ( F  e.  ( fBas `  B
)  ->  (/)  e/  F
)
7 df-nel 2645 . 2  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
86, 7sylib 196 1  |-  ( F  e.  ( fBas `  B
)  ->  -.  (/)  e.  F
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2642    e/ wnel 2643   A.wral 2793    i^i cin 3422    C_ wss 3423   (/)c0 3732   ~Pcpw 3955   dom cdm 4935   ` cfv 5513   fBascfbas 17910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fv 5521  df-fbas 17920
This theorem is referenced by:  fbdmn0  19520  fbncp  19525  fbun  19526  fbfinnfr  19527  0nelfil  19535  fsubbas  19553  fbasfip  19554  fgcl  19564  fbasrn  19570  uzfbas  19584  ucnextcn  19992
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