Theorem ntrss3

Description: The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007)

	Ref	Expression
Hypothesis	clscld.1	$$⊢X=\bigcup J$$
Assertion	ntrss3	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{int}\left(J\right)\left(S\right)\subseteq X$$

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	eltopss	$$⊢\left(J\in \mathrm{Top}\wedge \mathrm{int}\left(J\right)\left(S\right)\in J\right)\to \mathrm{int}\left(J\right)\left(S\right)\subseteq X$$
4	2 3	syldan	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq X\right)\to \mathrm{int}\left(J\right)\left(S\right)\subseteq X$$