Theorem isopn2

Description: A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006)

	Ref	Expression
Hypothesis	iscld.1	⊢ 𝑋 = ∪ 𝐽
Assertion	isopn2	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ 𝐽 ↔ ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ) )

Proof

Step	Hyp	Ref	Expression
1		iscld.1	⊢ 𝑋 = ∪ 𝐽
2		difss	⊢ ( 𝑋 ∖ 𝑆 ) ⊆ 𝑋
3	1	iscld2	⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑋 ∖ 𝑆 ) ⊆ 𝑋 ) → ( ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ∈ 𝐽 ) )
4	2 3	mpan2	⊢ ( 𝐽 ∈ Top → ( ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ∈ 𝐽 ) )
5		dfss4	⊢ ( 𝑆 ⊆ 𝑋 ↔ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) = 𝑆 )
6	5	biimpi	⊢ ( 𝑆 ⊆ 𝑋 → ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) = 𝑆 )
7	6	eleq1d	⊢ ( 𝑆 ⊆ 𝑋 → ( ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ∈ 𝐽 ↔ 𝑆 ∈ 𝐽 ) )
8	4 7	sylan9bb	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ↔ 𝑆 ∈ 𝐽 ) )
9	8	bicomd	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ 𝐽 ↔ ( 𝑋 ∖ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ) )