How do you show that two sets are disjoint? Here's a problem I am trying to solve for recreation. 
$$
A\cap B\subset C' \text{ and } A\cup C\subset B. \hspace{2 mm}\text{    Show that $A$ and $C$ are disjoint.}
$$
I can clearly see how A and C would be disjoint. Essentially, if my understanding is correct, A and C are non-overlapping sets within the bound of set B. But, I'm not exactly clear on how I would prove this by set logic. 
If you could provide some guidance, I would really appreciate it. 

$\def\wi\{\subseteq}$$A,C \wi B$ by the second condition and hence $A \cap C = ( A \cap B ) \cap C \wi C' \cap C = \varnothing$ by the first.
But if you want to prove from first principles: Take any $x \in A \cap C$. Then $x \in A$, so $x \in A \cup C$, and hence $x \in B$. Thus $x \in A \cap B$, so $x \in C'$. But we also have $x \in C$. Therefore there is no such $x$.
