Let each of $A, B$, and $C$ be a set and suppose $A \subseteq B \cup C$. Prove that $A \cap B \cap C = \varnothing$.
I start this problem by letting $x$ be an element of $A \subseteq B \cup C$ and stating that $x$ is an element of $A$ and also $B \cup C$. After that I get confused. Could someone provide some hints?