# Set Difference is Set

## 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

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.

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

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

Therefore:

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

$\blacksquare$