Theorem lpss

Description: The limit points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007)

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

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	$$⊢X=\bigcup J$$
2	1	lpsscls	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{limPt}\left(J\right)\left(S\right)\subseteq \mathrm{cls}\left(J\right)\left(S\right)$$
3	1	clsss3	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{cls}\left(J\right)\left(S\right)\subseteq X$$
4	2 3	sstrd	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{limPt}\left(J\right)\left(S\right)\subseteq X$$