Let $\Sigma_1$ and $\Sigma_2$ be sets of $L$-sentences such that no symbol of $L$ occurs in both $\Sigma_1$ and $\Sigma_2$. Suppose $\Sigma_1$ and $\Sigma_2$ have infinite models. Then $\Sigma_1 \cup \Sigma_2$ has a model.
We have some facts about back and forth systems and Vaught's test, and Löwenheim-Skolem's theorem, and by that last one I can say that $\Sigma_1, \Sigma_2$ have models of cardinality equal to the cardinality of $L$, but I don't know where to go from there. For instance, we don't necessarily want to show that $\Sigma_1 \cup \Sigma_2$ is complete, so things like Vaught's test seem irrelevant.
Does this just follow from a slick argument about compactness?