Union with Superset is Superset

Theorem

 * $S \subseteq T \iff S \cup T = T$

where:
 * $S \subseteq T$ denotes that $S$ is a subset of $T$;
 * $S \cup T$ denotes the union of $S$ and $T$.

Proof
Let $S \cup T = T$.

Then from the definition of set equality, $S \cup T \subseteq T$.

Thus:

Now let $S \subseteq T$.

From Subset of Union, we have $S \cup T \supseteq T$.

We also have:

So as we have:

it follows from the definition of Set Equality that we have $S \cup T = T$.

So we have:

and so:
 * $S \subseteq T \iff S \cup T = T$

from the definition of equivalence.

Also see

 * Intersection with Subset is Subset