Theorem opnneip

Description: An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007)

	Ref	Expression
Assertion	opnneip	$$⊢\left(J\in \mathrm{Top}\wedge N\in J\wedge P\in N\right)\to N\in \mathrm{nei}\left(J\right)\left(\left\{P\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		snssi	$$⊢P\in N\to \left\{P\right\}\subseteq N$$
2		opnneiss	$$⊢\left(J\in \mathrm{Top}\wedge N\in J\wedge \left\{P\right\}\subseteq N\right)\to N\in \mathrm{nei}\left(J\right)\left(\left\{P\right\}\right)$$
3	1 2	syl3an3	$$⊢\left(J\in \mathrm{Top}\wedge N\in J\wedge P\in N\right)\to N\in \mathrm{nei}\left(J\right)\left(\left\{P\right\}\right)$$