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Metamath Proof Explorer

Theorem sncld

Description: A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007) (Revised by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypothesis t1sep.1 ⊢ 𝑋 = ∪ 𝐽
Assertion sncld ( ( 𝐽 ∈ Haus ∧ 𝑃 ∈ 𝑋 ) → { 𝑃 } ∈ ( Clsd ‘ 𝐽 ) )

Proof

Step Hyp Ref Expression
1 t1sep.1 ⊢ 𝑋 = ∪ 𝐽
2 haust1 ⊢ ( 𝐽 ∈ Haus → 𝐽 ∈ Fre )
3 1 t1sncld ⊢ ( ( 𝐽 ∈ Fre ∧ 𝑃 ∈ 𝑋 ) → { 𝑃 } ∈ ( Clsd ‘ 𝐽 ) )
4 2 3 sylan ⊢ ( ( 𝐽 ∈ Haus ∧ 𝑃 ∈ 𝑋 ) → { 𝑃 } ∈ ( Clsd ‘ 𝐽 ) )