Question about the "pointed" property of proper cone

Question

I'm reading Boyd's Convex Optimization and have some question about the proper cone. The following is the property of a proper cone:
•K is convex.
• K is closed.
• K is solid, which means it has nonempty interior.
• K is pointed, which means that it contains no line (or equivalently, x ∈ K, − x ∈ K =⇒ x = 0).
How can "x ∈ K, − x ∈ K =⇒ x = 0" be proved? Supposed the vertex of the cone is not located at original point, then some point in the cone will not satisfy this property.