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.

Let $X = \map \complement S$.

Then:

$\ds X \subseteq T$

$\iff$

$\ds \map \complement X \cup T = \mathbb U$

$\ds \leadsto \ \ $

$\ds \map \complement S \subseteq T$

$\iff$

$\ds \map \complement {\map \complement S} \cup T = \mathbb U$

substituting $X = \map \complement S$

$\ds \leadsto \ \ $

$\ds \map \complement S \subseteq T$

$\iff$

$\ds S \cup T = \mathbb U$

$\blacksquare$

1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $11 \ \text{(a)}$
1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 3$: Set Operations: Union, Intersection and Complement: Exercise $1 \ \text{(d)}$