Superset of Unsatisfiable Set is Unsatisfiable

Theorem

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $\FF$ be an $\mathscr M$-unsatisfiable set of formulas from $\LL$.

Let $\FF'$ be a superset of $\FF$.

Then $\FF'$ is also $\mathscr M$-unsatisfiable.

Proof

By assumption, $\FF$ is unsatisfiable.

Suppose now $\FF'$ were satisfiable.

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

We conclude that $\FF'$ must be unsatisfiable.

$\blacksquare$