Set Complement inverts Subsets

Theorem
Let $$A \subseteq B \subseteq S$$.

Then:
 * $$\complement_S \left({B}\right) \subseteq \complement_S \left({A}\right)$$

where $$\complement_S$$ denotes the relative complement.

If $$S = \mathbb U$$ is taken as the universe, we can write it:


 * $$\complement \left({B}\right) \subseteq \complement \left({A}\right)$$

where $$\complement$$ denotes the set complement.

Proof
$$ $$ $$ $$

So:
 * $$A \subseteq B \iff \complement_S \left({B}\right) \subseteq \complement_S \left({A}\right)$$.

Setting $$S = \mathbb U$$ provides the result:
 * $$\complement \left({B}\right) \subseteq \complement \left({A}\right)$$