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 )