Consider two arbitrary subsets of $\mathbb R^n$, say $A$ and $B$ that aren't necessarily connected (here, I mean connected as in interligated, not in a mathematical sense) in any way possible.
Usually, when one wants to prove that $A \subset B$, what one does is to pick an arbitrary element $a \in A$ and shows that $a \in B$. My question is very simple: can we also prove that $A \subset B$ by proving that for any element $x \notin B$, it follows that $x \notin A?$
My intuition tells me that the answer should be yes, since I believe this a simple way to prove $A \subset B$ by contrapositive.
Thanks for any help in advance.