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