By binomial expansion, we have $$\begin{array}{rl} (1+m)^n =& \displaystyle\binom{n}{0}m^0 + \binom{n}{1}m^1 +\cdots + \binom{n}{n}m^n \\ (1+n)^m=& \displaystyle\binom{m}{0}n^0 + \binom{m}{1}n^1+ \cdots + \binom{m}{m}n^m \end{array}$$

The two first terms on the right of these two equations are equal. The next $m$ terms in the first expansion are all greater than their corresponding terms in the second expansion by the conclusion of # 4288.

Afterwards, there are no more team in the second expansion, and all the remaining terms in the first expansion are positive. Hence, the value of first expansion must be greater than that of the second expansion. This means the claim holds.