Finite Main Lemma of Propositional Tableaus

Lemma
Let $$\mathbf H$$ be a finite set of propositional WFFs.

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

Proof
Let $$\mathbf H$$ be a finite set of propositional WFFs which does not have a tableau confutation.

By the Tableau Extension Lemma, the tableau which consists only of a root node with hypothesis set $$\mathbf H$$ can be extended into a finite finished tableau $$T$$.

The tableau $$T$$ still has root $$\mathbf H$$.

Since $$T$$ is not a confutation, it has a finished branch $$\Gamma$$.

By the Finished Set Lemma, the set $$\Delta$$ of all WFFs on $$\Gamma$$ has a model, $$\mathcal M$$, say.

In particular, $$\mathcal M$$ is a model of $$\mathbf H$$ as required.

Comment
From Tableau Confutation means No Model, we already know that $$\mathbf H$$ can not have both a tableau confutation and a model.

This result gives us that $$\mathbf H$$ has a tableau confutation iff $$\mathbf H$$ does not have a model.