Compactness Theorem for Boolean Interpretations

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

Suppose every finite subset of $$\mathbf{H}$$ has a model.

Then $$\mathbf{H}$$ has a model.

Proof
Suppose $$\mathbf{H}$$ does not have a model.

By the Main Lemma of Propositional Calculus, $$\mathbf{H}$$ has a tableau confutation $$T$$.

Since each tableau confutation is a finite tableau, the set $$\mathbf{H}'$$ of all propositional WFFs in $$\mathbf{H}$$ used somewhere in $$T$$ is finite.

Now, let $$T'$$ be the labeled tree which is the same as $$T$$ but with root $$\mathbf{H}'$$ instead of $$\mathbf{H}$$.

Then $$T'$$ is a tableau confutation of $$\mathbf{H}'$$.

By the Extended Soundness Theorem of Propositional Calculus, $$\mathbf{H}'$$ has no models.

But this contradicts the assumption that all finite subsets of $$\mathbf{H}$$ have models.

Hence the result.