Suppose $A \subseteq C$ and $B$ and $C$ are disjoint. Prove that $x \in A \rightarrow x \notin B$.
Basically I need to prove this.
In proving these things you need to show that no matter how $A,B,C$ look like, if the condition that $A\subseteq C$ and $B\cap C=\varnothing$ then $x\in A\rightarrow x\notin B$ is true.
For this we want to take an arbitrary element of $A$, use the definition that $A\subseteq C$ to deduce more about $x$; and use the definition of $B\cap C$ to deduce that if $x$ was in $B$ then the intersection would not be empty - therefore $x\notin B$.
I leave the formal and technical details to you, since this is your homework assignment after all.