[Math] How to show that two sets are disjoint

Question

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.

Answer 1

The second condition says that all of $A$ is inside $B$. So $A\cap B = A\subset C'$.