Theorem 0nelfb 20064

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.

Step	Hyp	Ref	Expression
1		elfvdm 5890	. . . . 5 |- ( F e. ( fBas ` B ) -> B e. dom fBas )
2		isfbas 20062	. . . . 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 ) ) =/= (/) ) ) ) )
3	1, 2	syl 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 ) ) =/= (/) ) ) ) )
4	3	ibi 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 1003	. . 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 )
6	4, 5	syl 16	. 2 |- ( F e. ( fBas ` B ) -> (/) e/ F )
7		df-nel 2665	. 2 |- ( (/) e/ F <-> -. (/) e. F )
8	6, 7	sylib 196	1 |- ( F e. ( fBas ` B ) -> -. (/) e. F )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   e. wcel 1767   =/= wne 2662   e/ wnel 2663   A.wral 2814   i^i cin 3475   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   dom cdm 4999   ` cfv 5586   fBascfbas 18174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-fbas 18184

This theorem is referenced by:  fbdmn0  20067  fbncp  20072  fbun  20073  fbfinnfr  20074  0nelfil  20082  fsubbas  20100  fbasfip  20101  fgcl  20111  fbasrn  20117  uzfbas  20131  ucnextcn  20539