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Metamath Proof Explorer

Theorem mopni2

Description: An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Hypothesis mopni.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion mopni2 ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 )

Proof

Step Hyp Ref Expression
1 mopni.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2 1 mopni ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) )
3 1 mopnss ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝑋 )
4 3 sselda ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ 𝑋 )
5 blssex ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ( ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 ) )
6 5 adantlr ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) ∧ 𝑃 ∈ 𝑋 ) → ( ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 ) )
7 4 6 syldan ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) ∧ 𝑃 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 ) )
8 7 3impa ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 ) )
9 2 8 mpbid ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 )