Is there a reference for this ZF theorem?

Let $m,n\in\mathbb{N}$. If $a_1,\dots,a_m$ and $b_1,\dots,b_n$ are cardinals such that $a_i\le b_j$ for all $i$ and $j$, then there is a cardinal $x$ such that $a_i\le x\le b_j$ for all $i$ and $j$.

It's enough if the proposition is stated for the case $m = n = 2$, as the rest follows by an easy induction.

Is there a published reference for this ZF theorem?

Let $m,n\in\mathbb{N}$. If $a_1,\dots,a_m$ and $b_1,\dots,b_n$ are cardinals such that $a_i\le b_j$ for all $i$ and $j$, then there is a cardinal $x$ such that $a_i\le x\le b_j$ for all $i$ and $j$.

It's enough if the proposition is stated for the case $m = n = 2$, as the rest follows by an easy induction.