proving unsatisfiability in a union of closed WFF

Question

If I am given a closed set of wff $X$ and it is unsatisfiable, then how do I show that the set $X \cup \{A\}$, where $A$ is any closed wff, is unsatisfiable?

Answer 1

Let $X$ be unsatisfiable. Then, by compactness, there is a finite subset $Y$ of $X$ that is unsatisfiable. Since $Y \subseteq X \cup \{A\}$, then it follows that $X \cup \{A\}$ is also unsatisfiable.