Metamath Proof Explorer

Theorem bln0

Description: A ball is not empty. (Contributed by NM, 6-Oct-2007) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Assertion bln0 
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> ( P ( ball ` D ) R ) =/= (/) )

Proof

Step Hyp Ref Expression
1 blcntr 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> P e. ( P ( ball ` D ) R ) )
2 1 ne0d 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> ( P ( ball ` D ) R ) =/= (/) )