Theorem iscld2

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

	Ref	Expression
Hypothesis	iscld.1	⊢ 𝑋 = ∪ 𝐽
Assertion	iscld2	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑋 ∖ 𝑆 ) ∈ 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		iscld.1	⊢ 𝑋 = ∪ 𝐽
2	1	iscld	⊢ ( 𝐽 ∈ Top → ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑆 ⊆ 𝑋 ∧ ( 𝑋 ∖ 𝑆 ) ∈ 𝐽 ) ) )
3	2	baibd	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑋 ∖ 𝑆 ) ∈ 𝐽 ) )