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 $$\Gamma$$, $$\mathcal M \models \mathbf H$$.