Set of Logical Formulas is Inconsistent iff it has Finite Inconsistent Subset

Theorem
Let $\FF$ be a collection of logical formulas.

Then:
 * $\FF$ be inconsistent

there exists a finite subset of $\FF$ which it itself inconsistent.

Sufficient Condition
Let $\FF$ be inconsistent.

Then it is possible to assemble a proof in a finite set of statements of a contradiction.

This finite set of statements uses within it a finite subset $\GG \subseteq \FF$ of the logical formulas of $\FF$.

Hence $\GG$ is that inconsistent finite subset of $\FF$ whose existence is proposed.

Necessary Condition
Let $\FF$ have a finite subset $\GG$ which is inconsistent.

Then it is possible to assemble a proof of a contradiction using logical formulas of $\GG$.

But those logical formulas are also logical formulas of $\FF$.

Then by definition $\FF$ is likewise inconsistent.