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 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		blcntr	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → 𝑃 ∈ ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) )
2	1	ne0d	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+ ) → ( 𝑃 ( ball ‘ 𝐷 ) 𝑅 ) ≠ ∅ )