Enderton Problem 6 in Section 2.5

Let and be sets of sentences such that nothing is a model of both and . Show that there is a sentence such that and . (This can be stated Disjoint classes can be separated by an EC class.) Suggestion: is unsatisfiable; apply compactness.

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Since is unsatisfiable, it is inconsistent by the completeness theorem. Therefore, and . I am stuck here.

Any help will be highly appreciated.