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.