# Unsatisfiable Set Union Formula 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 $\phi$ be a logical formula.

Then $\mathcal F \cup \left\{{\phi}\right\}$ is also $\mathscr M$-unsatisfiable.

## Proof

This is an immediate consequence of Superset of Unsatisfiable Set is Unsatisfiable.

$\blacksquare$