Union with Superclass is Superclass

Theorem
Let $A$ and $B$ be classes.

Then:
 * $A \subseteq B \iff A \cup B = B$

where:
 * $A \subseteq B$ denotes that $A$ is a subclass of $B$
 * $A \cup B$ denotes the union of $A$ and $B$.

Proof
Let $A \cup B = B$.

Then by definition of class equality:
 * $A \cup B \subseteq B$

Thus:

Now let $A \subseteq B$.

We have:

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

We also have:

Thus:

By definition of class equality:
 * $A \cup B = B$

Also see

 * Intersection with Subclass is Subclass