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