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

Then $\FF \cup \set \phi$ is also $\mathscr M$-unsatisfiable.

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

Also see

 * Superset of Unsatisfiable Set is Unsatisfiable