Theorem cmntrcld

Description: The complement of an interior is closed. (Contributed by NM, 1-Oct-2007) (Proof shortened by OpenAI, 3-Jul-2020)

	Ref	Expression
Hypothesis	clscld.1	⊢ 𝑋 = ∪ 𝐽
Assertion	cmntrcld	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) ∈ ( Clsd ‘ 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	⊢ 𝑋 = ∪ 𝐽
2	1	ntropn	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ∈ 𝐽 )
3	1	opncld	⊢ ( ( 𝐽 ∈ Top ∧ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ∈ 𝐽 ) → ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) ∈ ( Clsd ‘ 𝐽 ) )
4	2 3	syldan	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) ∈ ( Clsd ‘ 𝐽 ) )