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