For all ordinals $\alpha$, $\beta$, and $\gamma$ with $\beta \leq \gamma$, the following inequalities hold:
\begin{align*}
\alpha + \beta &\leq \alpha + \gamma,\\
\alpha \cdot \beta &\leq \alpha \cdot \gamma.
\end{align*}
Moreover, if $\alpha > 0$, then
\begin{align*}
\alpha^\beta \leq \alpha^\gamma.
\end{align*}