Prove that $$A\cap B=\emptyset\land C\subseteq A\cup\overline{B}\to C\subseteq\overline{B}$$ using set operations.
My proof:
\begin{align*} C&\subseteq A\cup\overline{B}\\ &=(A\cup\overline{B})\cap(B\cup\overline{B})\\ &=(A\cap B)\cup(A\cap\overline{B})\cup(\overline{B}\cap B)\cup(\overline{B}\cap\overline{B})\\ &=\emptyset\cup(A\cap\overline{B})\cup\emptyset\cup\overline{B}\\ &=\overline{B}\cup(A\cap\overline{B})\\ &=\overline{B}. \end{align*}
Therefore $C\subseteq\overline{B}$.
Is my proof correct?