Exponential on Real Numbers is Injection

Theorem
Let $\exp: \R \to \R$ be the exponential function:
 * $\map \exp x = e^x$

Then $\exp$ is an injection.

Proof
From Exponential is Strictly Increasing:
 * $\exp$ is strictly increasing on $\R$.

From Strictly Monotone Mapping with Totally Ordered Domain is Injective:
 * $\exp$ is an injection.