Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma.
I've a Lemma saying:
Suppose $P=P(V,E)$ where $V,E \in \mathbb R^n$ are finite sets. If $0_n$ (zero) is not an extreme point of $\text {cone}(E)$, then $P$ has no extreme points.
I should then look at when $P$ contain a line vs. when $0_n$ is an extreme point of $\text{cone}(E)$.
I've found the following:
$$\exists \ \text {extreme point} \ x \in P \Rightarrow 0_n \ \text {is an extreme point of cone}(E)$$
$$0_n \ \text {is an extreme point of cone}(E) \Rightarrow \forall \ v \in V \ (v\text { is an extreme point of } P)$$
$$0_n \ \text {is an extreme point of cone}(E) \Rightarrow \forall \ e \in E \ (e\text { is an extreme direction of } P)$$
However, I cannot conclude the above statement from this.
Help is appreciated !