Class Difference with Class Difference with Subclass
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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
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 $5.6. \ \text {(g)}$