Let A and B be two non-empty subsets of set X such that A is not a subset of B, then

If $$A$$ is not a subset of $$B$$, there must exist at least one element $$x$$ that belongs to $$A$$ but does not belong to $$B$$. By definition, any element not in $$B$$ must be in its complement, $$B^c$$, which means $$x$$ is common to both $$A$$ and $$B^c$$. Since their intersection contains at least this one element ($$A \cap B^c \neq \emptyset$$), the two sets are "non-disjoint" by necessity.