Enderton Problem 6 in Section 2.5

Let $\displaystyle \Sigma_1$ and $\displaystyle \Sigma_2$ be sets of sentences such that nothing is a model of both $\displaystyle \Sigma_1$ and $\displaystyle \Sigma_2$. Show that there is a sentence $\displaystyle \tau$ such that $\displaystyle \text{Mod } \Sigma_1 \subseteq \text{Mod }\tau$ and $\displaystyle \text{Mod } \Sigma_2 \subseteq \text {Mod } \neg \tau$. (This can be stated Disjoint $\displaystyle \text{EC}_\Delta $ classes can be separated by an EC class.) Suggestion: $\displaystyle \Sigma_1 \cup \Sigma_2$ is unsatisfiable; apply compactness.

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Since $\displaystyle \Sigma_1 \cup \Sigma_2$ is unsatisfiable, it is inconsistent by the completeness theorem. Therefore, $\displaystyle \Sigma_1 \cup \Sigma_2 \vdash \tau$ and $\displaystyle \Sigma_1 \cup \Sigma_2 \vdash \neg \tau$. I am stuck here.

Any help will be highly appreciated.