Complement Union with Superset is Universe/Corollary

Corollary to Complement Union with Superset is Universe

 * $S \cup T = \mathbb U \iff \map \complement S \subseteq T$

where:
 * $S \subseteq T$ denotes that $S$ is a subset of $T$
 * $S \cup T$ denotes the union of $S$ and $T$
 * $\complement$ denotes set complement
 * $\mathbb U$ denotes the universal set.

Proof
Let $X = \map \complement S$.

Then:

Also see

 * Empty Intersection iff Subset of Complement