Barwise compactness theorem

In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967.

Statement
Let $$A$$ be a countable admissible set. Let $$L$$ be an $$A$$-finite relational language. Suppose $$\Gamma$$ is a set of $$L_A$$-sentences, where $$\Gamma$$ is a $$\Sigma_1$$ set with parameters from $$A$$, and every $$A$$-finite subset of $$\Gamma$$ is satisfiable. Then $$\Gamma$$ is satisfiable.