Finished Set Lemma/Corollary

Corollary to Finished Set Lemma
Let $\Delta$ be a finished set of WFFs of propositional logic.

Then $\Delta$ has a model.

Proof
Note that the set of basic WFFs in $\Delta$ has at least one model.

Let $v$ be the boolean interpretation defined as follows:
 * $v (p) = \begin{cases}

T & : p \in \Delta \\ F & : p \notin \Delta \end{cases}$

Because $\Delta$ is finished, it is not contradictory, and hence $v (p) = F$ if $\neg p \in \Delta$.

Thus $v$ is a model of the basic WFFs of $\Delta$.

The result follows by the Finished Set Lemma.