Union with Superset is Superset/Proof 1

Proof
Let $S \cup T = T$.

Then by 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:

Then:

By definition of set equality:
 * $S \cup T = T$

So:

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

from the definition of equivalence.