Corollary 2

If $\mathfrak{a}$, $\mathfrak{a}'$, $\mathfrak{b}$, $\mathfrak{b}'$ are cardinals such that $\mathfrak{a} \leqslant \mathfrak{a}'$, $\mathfrak{b} \leqslant \mathfrak{b}'$, and $\mathfrak{a}' > 0$, then $\mathfrak{a}^\mathfrak{b} \leqslant \mathfrak{{a}'}^{\mathfrak{b}'}$.

For $\mathfrak{a}^\mathfrak{b} \leqslant \mathfrak{{a}'}^\mathfrak{b}$ by Propositions 10 and 14, and $\mathfrak{{a}'}^\mathfrak{b} \leqslant \mathfrak{{a}'}^{\mathfrak{b}'}$ by Proposition 10 and Corollary 1 to Proposition 14.