Theorem cldopn

Description: The complement of a closed set is open. (Contributed by NM, 5-Oct-2006) (Revised by Stefan O'Rear, 22-Feb-2015)

	Ref	Expression
Hypothesis	iscld.1	⊢ 𝑋 = ∪ 𝐽
Assertion	cldopn	⊢ ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) → ( 𝑋 ∖ 𝑆 ) ∈ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		iscld.1	⊢ 𝑋 = ∪ 𝐽
2		cldrcl	⊢ ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) → 𝐽 ∈ Top )
3	1	iscld	⊢ ( 𝐽 ∈ Top → ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) ↔ ( 𝑆 ⊆ 𝑋 ∧ ( 𝑋 ∖ 𝑆 ) ∈ 𝐽 ) ) )
4	3	simplbda	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ ( Clsd ‘ 𝐽 ) ) → ( 𝑋 ∖ 𝑆 ) ∈ 𝐽 )
5	2 4	mpancom	⊢ ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) → ( 𝑋 ∖ 𝑆 ) ∈ 𝐽 )