# Set Difference is Set

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## Theorem

Let $x$ be a small class.

Let $A$ be a class.

Let $\map \MM B$ denote that $B$ is small.

Then:

- $\map \MM {x \setminus A}$

## Proof

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By Set Difference as Intersection with Relative Complement:

- $\paren {x \setminus A} = \paren {x \cap \map \complement A}$

Next, by Axiom of Subsets Equivalents, $\paren {x \cap \map \complement A}$ is small.

Therefore:

- $\map \MM {x \setminus A}$

$\blacksquare$

## Sources

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