# Set Difference is Set

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## Theorem

Let $x$ be a small class.

Let $A$ be a class.

Let $\mathcal M \left({B}\right)$ denote that $B$ is small.

Then:

- $\mathcal M \left({x \setminus A}\right)$

## Proof

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By Set Difference as Intersection with Relative Complement:

- $\left({x \setminus A}\right) = \left({x \cap \complement \left({A}\right)}\right)$

Next, by Axiom of Subsets Equivalents, $\left({x \cap \complement \left({A}\right)}\right)$ is small.

Therefore:

- $\mathcal M \left({x \setminus A}\right)$

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 5.15$