We have established the following result regarding the Dushnik–Miller dimension of posets.

Let $P$ be a poset with downsets $C, D \subseteq P$. If the dimensions of $C$ and $D$ are $m$ and $n$, respectively, then the dimension of $C \cup D$ is at most $m+n$.

The proof is elementary, but we've been unable to find a reference for this. Is anyone aware of a reference for this or a more general result?