Assume $A$, $B$ are included in a common superset $D$. Verify the identity $A^c \subseteq B^c \iff B^c \subseteq A^c$
Incomplete Proof: Suppose $A^c \subseteq B^c$ and pick $x \in B^c$. We need to show $x \in A^c$
This is a problem I'm really stuck on. I'm not sure how to proceed with a direct proof. If I try to argue by contradiction and suppose that $x \in A$ I still get nowehere and if I try and use the contrpositive and show that $B^c \not\subseteq A^c \implies A^c \not\subseteq B^c$, then there'd exist an $x \in B^c$ for which $x \in A$, but again I feel like I don't have enough information to show the existence of a $y \in A^c$ such that $y \not\in B^c$
How could I go about proving this?