Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas

Theorem
Let $\mathscr G$ be instance 1 of a Gentzen proof system. Let $U$ be a set of WFFs of propositional logic.

Then $U$ is a $\mathscr G$-theorem iff:


 * $\neg U$ has a closed semantic tableau

where $\neg U = \left\{{\neg \mathbf A: \mathbf A \in U}\right\}$.