Theorem lpss2

Description: Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Hypothesis	lpss2.1	$$⊢X=\bigcup J$$
Assertion	lpss2	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq X\wedge B\subseteq A\right)\to \mathrm{limPt}\left(J\right)\left(B\right)\subseteq \mathrm{limPt}\left(J\right)\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		lpss2.1	$$⊢X=\bigcup J$$
2	1	lpss3	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq X\wedge B\subseteq A\right)\to \mathrm{limPt}\left(J\right)\left(B\right)\subseteq \mathrm{limPt}\left(J\right)\left(A\right)$$