# Intersection of Class and Set is Set

## Theorem

Let $C$ be the class:

$C = \set { u : \map \phi {u, p_1, \ldots, p_n} }$

Then for all sets $X$, $C \cap X$ is a set.

## Proof

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By the definition of class intersection:

$a \in C \cap X \implies a \in C \land a \in X$

Thus:

$a \in C \cap X \implies a \in X$

The subclass definition gives:

$C \cap X \subseteq X$

By Subclass of Set is Set, $C \cap X$ is a set.

$\blacksquare$