Main Lemma of Propositional Tableaus

Lemma
Let $$\mathbf{H}$$ be a countable set of propositional WFFs.

Either $$\mathbf{H}$$ has a tableau confutation or $$\mathbf{H}$$ has a model.

Proof
If $$\mathbf{H}$$ is finite, then the Finite Main Lemma applies.

So, assume that $$\mathbf{H}$$ is countably infinite.

Suppose $$\mathbf{H}$$ does not have a tableau confutation.

Because a Countable Hypothesis Set has Finished Tableau, there is a finished tableau $$T$$ with hypothesis set $$\mathbf{H}$$.

By Finished Tableau has Finished Branch or is a Confutation‎, as $$T$$ is (by hypothesis) not a confutation, it must have a finished branch; call it $$\Gamma$$.

By the Finished Set Lemma, the set of propositional WFFs on $$\Gamma$$ has a model $$\mathcal{M}$$.

Finally, because all the WFFs in the hypothesis set occur on $$\Gamma$$, $$\mathcal{M}\models \mathbf{H}$$.