Theorem clsss3

Description: The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007)

	Ref	Expression
Hypothesis	clscld.1	$$⊢X=\bigcup J$$
Assertion	clsss3	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{cls}\left(J\right)\left(S\right)\subseteq X$$

Proof

Step	Hyp	Ref	Expression
1		clscld.1	$$⊢X=\bigcup J$$
2	1	clscld	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{cls}\left(J\right)\left(S\right)\in \mathrm{Clsd}\left(J\right)$$
3	1	cldss	$$⊢\mathrm{cls}\left(J\right)\left(S\right)\in \mathrm{Clsd}\left(J\right)\to \mathrm{cls}\left(J\right)\left(S\right)\subseteq X$$
4	2 3	syl	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{cls}\left(J\right)\left(S\right)\subseteq X$$