# Intersection of Class and Set is Set

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## 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.

$\blacksquare$

## Sources

- 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Separation Schema