Prof. Theodore Shifrin wrote "a differentiable function on an interval in $\mathbb{R}$ with nowhere zero drivative has a differentiable inverse"

Question

Prof. Theodore Shifrin wrote "a differentiable function on an interval in $\mathbb{R}$ with nowhere zero drivative has a differentiable inverse" on p.251 in his "Multivariable Mathematics".

Is the following argument ok?

If $f$ is differentiable on $I$ and $f^{\prime}(x)\ne 0$ on $I$, then by the intermediate value theorem, $f^{\prime}(x) > 0$ on $I$ or $f^{\prime}(x) < 0$ on $I$. So $f$ is strictly increasing on $I$ or $f$ is strictly decreasing on $I$. So $f$ has a differentiable inverse.