No open subset $U \subset \mathbb{C}$ so that $f|_U :U \rightarrow \mathbb{C} \smallsetminus {0}$ where $f=e^z$ is a bijection

Question

I nood to show that there is no open subset $U \subset \mathbb{C}$ so that $f|_U :U \rightarrow \mathbb{C} \smallsetminus {0}$ where $f=e^z$ is a bijection.

I started by claiming that for $f$ to be $1-1$, $arg(z)$ must be in the set $[\alpha +2 \pi k,\alpha +4 \pi k)$ for all $z \in U$ for a particular $\alpha \in [0,2 \pi)$ and $k \in \mathbb{Z}$. Hence $U$ is contained by a "vertical strip" on the plane, I get nowhere from here.