Superset of Unsatisfiable Set is Unsatisfiable
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.
By assumption, $\mathcal F$ is unsatisfiable.
Suppose now $\mathcal F'$ were satisfiable.
We conclude that $\mathcal F'$ must be unsatisfiable.