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Metamath Proof Explorer
Description: The intersection of two open sets of a metric space is open.
(Contributed by NM, 4-Sep-2006) (Revised by Mario Carneiro, 23-Dec-2013)
|
|
Ref |
Expression |
|
Hypothesis |
mopni.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
|
Assertion |
mopnin |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝐽 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mopni.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 2 |
1
|
mopntop |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top ) |
| 3 |
|
inopn |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝐽 ) |
| 4 |
2 3
|
syl3an1 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝐽 ) |