Suppose that A is a subset of B. How can we show that B-complement is a subset of A-complement?
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Suppose $b\in B^c$. Is it possible that $b\in A$? Suppose it is, and derive a contradiction, and you should have your desired containment.
Instead of using a proof by contradiction, $b\in B^c \implies b \notin A$ since $A \subset B$ hence $b \in A^c$ which implies the result.
The key here being that for all $x$, either $x\in X$ or $x\in X^c$.