Extension of Contradictory Branch is Contradictory

Theorem

Let $T$ be a propositional tableau.

Let $\Gamma$ be a contradictory branch of $T$.

Let $\Gamma'$ be an extension of $\Gamma$.

Then $\Gamma'$ is also contradictory.

Proof

Since $\Gamma$ is contradictory, there is some WFF $\mathbf A$ such that both $\mathbf A$ and $\neg \mathbf A$ occur along $\Gamma$.

Since $\Gamma'$ is an extension of $\Gamma$, $\mathbf A$ and $\neg \mathbf A$ also occur along $\Gamma'$.

Hence $\Gamma'$ is contradictory.

$\blacksquare$