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Metamath Proof Explorer

Theorem sst1

Description: A topology finer than a T_1 topology is T_1. (Contributed by Mario Carneiro, 25-Aug-2015)

Ref Expression
Hypothesis t1sep.1 ⊢ 𝑋 = ∪ 𝐽
Assertion sst1 ( ( 𝐽 ∈ Fre ∧ 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) → 𝐾 ∈ Fre )

Proof

Step Hyp Ref Expression
1 t1sep.1 ⊢ 𝑋 = ∪ 𝐽
2 t1top ⊢ ( 𝐽 ∈ Fre → 𝐽 ∈ Top )
3 cnt1 ⊢ ( ( 𝐽 ∈ Fre ∧ ( I ↾ 𝑋 ) : 𝑋 –1-1→ 𝑋 ∧ ( I ↾ 𝑋 ) ∈ ( 𝐾 Cn 𝐽 ) ) → 𝐾 ∈ Fre )
4 1 2 3 sshauslem ⊢ ( ( 𝐽 ∈ Fre ∧ 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) → 𝐾 ∈ Fre )