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Theorem fbasne0 17815
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 5716 . . . 4  |-  ( F  e.  ( fBas `  B
)  ->  B  e.  dom  fBas )
2 isfbas 17814 . . . 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 233 . 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 963 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2567    e/ wnel 2568   A.wral 2666    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   dom cdm 4837   ` cfv 5413   fBascfbas 16644
This theorem is referenced by:  fbdmn0  17819  fbssint  17823  fbun  17825  trfbas2  17828  filtop  17840  fsubbas  17852  fgcl  17863  fbasrn  17869  fmfnfm  17943
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-fbas 16654
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