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	$$⊢X=\bigcup J$$
Assertion	cmntrcld	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to X\setminus \mathrm{int}\left(J\right)\left(S\right)\in \mathrm{Clsd}\left(J\right)$$

Proof

Step	Hyp	Ref	Expression
1		clscld.1	$$⊢X=\bigcup J$$
2	1	ntropn	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{int}\left(J\right)\left(S\right)\in J$$
3	1	opncld	$$⊢\left(J\in \mathrm{Top}\wedge \mathrm{int}\left(J\right)\left(S\right)\in J\right)\to X\setminus \mathrm{int}\left(J\right)\left(S\right)\in \mathrm{Clsd}\left(J\right)$$
4	2 3	syldan	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to X\setminus \mathrm{int}\left(J\right)\left(S\right)\in \mathrm{Clsd}\left(J\right)$$