Equivalence of Definitions of Logical Consistence

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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.