Intersection with Subclass is Subclass

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

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

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

Proof
Let $A \cap B = A$.

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

Thus:

Now let $A \subseteq B$.

We also have:

So as we have:

it follows from the definition of class equality that:
 * $A \cap B = A$

Also see

 * Union with Superclass is Superclass