Set Complement inverts Subsets/Proof 3

Theorem

$S \subseteq T \iff \map \complement T \subseteq \map \complement S$

Proof

By definition of set complement:

$\map \complement T := \relcomp {\mathbb U} T$

where:

$\mathbb U$ is the universe

$\relcomp {\mathbb U} T$ denotes the complement of $T$ relative to $\mathbb U$.

Thus the statement can be expressed as:

$S \subseteq T \iff \relcomp {\mathbb U} T \subseteq \relcomp {\mathbb U} S$

The result then follows from Relative Complement inverts Subsets.

$\blacksquare$