Set Complement inverts Subsets/Proof 3

Theorem

 * $S \subseteq T \iff \complement \left({T}\right) \subseteq \complement \left({S}\right)$

where:
 * $S \subseteq T$ denotes that $S$ is a subset of $T$
 * $\complement$ denotes set complement.

Proof
By definition of set complement:
 * $\complement \left({T}\right) := \complement_\mathbb U \left({T}\right)$

where:
 * $\mathbb U$ is the universe
 * $\complement_\mathbb U \left({T}\right)$ denotes the complement of $T$ relative to $\mathbb U$.

Thus the statement can be expressed as:
 * $S \subseteq T \iff \complement_\mathbb U \left({T}\right) \subseteq \complement_\mathbb U \left({S}\right)$

The result then follows from Relative Complements invert Subsets.