Consistency of Logical Formulas has Finite Character

Theorem
Let $P$ be the property of collections of logical formulas defined as:


 * $\forall \FF: \map P \FF$ denotes that $\FF$ is consistent.

Then $P$ is of finite character.

That is:
 * $\FF$ is a consistent set of formulas every finite subset of $\FF$ is also a consistent set of formulas.