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.

$\blacksquare$

Also see

• Relative Complement of Relative Complement

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom: Exercise $5.6. \ \text {(g)}$