# Superset of Unsatisfiable Set is Unsatisfiable

## Theorem

Let $\mathcal L$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\mathcal L$.

Let $\mathcal F$ be an $\mathscr M$-unsatisfiable set of formulas from $\mathcal L$.

Let $\mathcal F'$ be a superset of $\mathcal F$.

Then $\mathcal F'$ is also $\mathscr M$-unsatisfiable.

## Proof

By assumption, $\mathcal F$ is unsatisfiable.

Suppose now $\mathcal F'$ were satisfiable.

Then it would follow from Subset of Satisfiable Set is Satisfiable that $\mathcal F$ were also satisfiable.

We conclude that $\mathcal F'$ must be unsatisfiable.

$\blacksquare$