Let $A$ and $B$ be subsets of the universal set $U$. Prove that $$A^{c}\subseteq B\cap C^{c} \Longleftrightarrow (A\cup B)\cap(A\cup C^{c}) = U$$
We haven't learned any laws surrounding subsets in proofs, I feel like law of subtraction may be a good place to start but I can't figure out where to go