Finished Propositional Tableau has Finished Branch or is Confutation

Theorem
Let $\left({T, \mathbf H, \Phi}\right)$ be a finished propositional tableau.

Then one of the following holds:


 * $T$ has a finished branch
 * $T$ is a confutation.

Proof
Suppose $T$ has no finished branch.

Then every branch of $T$ is finite and contradictory.

In this case, since every branch of $T$ is finite, $T$ is a finite tableau by König's Tree Lemma‎.

Since, then, $T$ is finite and every branch of $T$ is contradictory, $T$ is a tableau confutation.