Metamath Proof Explorer

Theorem fbasne0

Description: There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009) (Revised by Mario Carneiro, 28-Jul-2015)

Ref Expression
Assertion fbasne0 
|- ( F e. ( fBas ` B ) -> F =/= (/) )

Proof

Step Hyp Ref Expression
1 elfvdm 
 |-  ( F e. ( fBas ` B ) -> B e. dom fBas )
2 isfbas 
 |-  ( 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 
 |-  ( 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 
 |-  ( 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 
 |-  ( ( F C_ ~P B /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) -> F =/= (/) )
6 4 5 syl 
 |-  ( F e. ( fBas ` B ) -> F =/= (/) )