Equivalence of Definitions of Logical Consistence

Theorem
Let $\mathbf H$ be a countable set (either finite or infinite) of WFFs of propositional logic.

The following statements are logically equivalent:


 * $(1): \quad$ $\mathbf H$ has a model.


 * $(2): \quad$ $\mathbf H$ is consistent for the proof system of propositional tableaus.


 * $(3): \quad$ $\mathbf H$ has no tableau confutation.