Relative Complement inverts Subsets

Theorem
Let $S$ be a set.

Let $A \subseteq S, B \subseteq S$ be subsets of $S$.

Then:


 * $A \subseteq B \iff \relcomp S B \subseteq \relcomp S A$

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

Also known as
This result can be referred to by saying that:
 * the subset operation is inclusion-inverting
 * the relative complement operation, considered as a mapping on the ordered structure $\struct {S, \subseteq}$, is decreasing