Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$. Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.
The forward implication is easy to prove but i got stuck at the backward implication, i.e. $(U\setminus B)\subseteq(U\setminus A)\Rightarrow A\subseteq B$. Can someone help me out with it? Thank you.

HINT: Let $C=U\setminus B$ and $D=U\setminus A$. The hypothesis that $U\setminus B\subseteq U\setminus A$ can then be written $C\subseteq D$. Now apply the forward direction to $C$ and $D$.
