Class Difference with Class Difference with Subclass

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

Let $B \subseteq A$.

Then:
 * $A \setminus \paren {A \setminus B} = B$

Proof
From Class Difference with Class Difference:
 * $A \setminus \paren {A \setminus B} = A \cap B$

for all classes $A$ and $B$.

From Intersection with Subclass is Subclass:


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

The result follows.

Also see

 * Relative Complement of Relative Complement