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