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Theorem 0nelfb 17578
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 5592 . . . . 5  |-  ( F  e.  ( fBas `  B
)  ->  B  e.  dom  fBas )
2 isfbas 17576 . . . . 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 15 . . . 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 232 . . 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 962 . . 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 15 . 2  |-  ( F  e.  ( fBas `  B
)  ->  (/)  e/  F
)
7 df-nel 2482 . 2  |-  ( (/)  e/  F  <->  -.  (/)  e.  F
)
86, 7sylib 188 1  |-  ( F  e.  ( fBas `  B
)  ->  -.  (/)  e.  F
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1701    =/= wne 2479    e/ wnel 2480   A.wral 2577    i^i cin 3185    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   dom cdm 4726   ` cfv 5292   fBascfbas 16421
This theorem is referenced by:  fbdmn0  17581  fbncp  17586  fbun  17587  fbfinnfr  17588  0nelfil  17596  fsubbas  17614  fbasfip  17615  fgcl  17625  fbasrn  17631  uzfbas  17645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fv 5300  df-fbas 16429
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