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Metamath Proof Explorer

Theorem sshaus

Description: A topology finer than a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015)

Ref Expression
Hypothesis t1sep.1 ⊢ 𝑋 = ∪ 𝐽
Assertion sshaus ( ( 𝐽 ∈ Haus ∧ 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) → 𝐾 ∈ Haus )

Proof

Step Hyp Ref Expression
1 t1sep.1 ⊢ 𝑋 = ∪ 𝐽
2 haustop ⊢ ( 𝐽 ∈ Haus → 𝐽 ∈ Top )
3 cnhaus ⊢ ( ( 𝐽 ∈ Haus ∧ ( I ↾ 𝑋 ) : 𝑋 –1-1→ 𝑋 ∧ ( I ↾ 𝑋 ) ∈ ( 𝐾 Cn 𝐽 ) ) → 𝐾 ∈ Haus )
4 1 2 3 sshauslem ⊢ ( ( 𝐽 ∈ Haus ∧ 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) → 𝐾 ∈ Haus )