This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006)
|
|
Ref |
Expression |
|
Hypothesis |
mopni.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
|
Assertion |
rnblopn |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐵 ∈ ran ( ball ‘ 𝐷 ) ) → 𝐵 ∈ 𝐽 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mopni.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 2 |
1
|
blssopn |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) ⊆ 𝐽 ) |
| 3 |
2
|
sselda |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐵 ∈ ran ( ball ‘ 𝐷 ) ) → 𝐵 ∈ 𝐽 ) |