Intersection of Non-Empty Class is Set/Proof 2
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Theorem
Let $\bigcap A$ denote the intersection of $A$.
Then $\bigcap A$ is a set.
Proof
Since $A$ is a non-empty class, there exists $S \in A$.
Since $S$ is an element of a class, it is not a proper class, and is thus a set.
By definition of class intersection:
- $x \in \bigcap A \implies x \in S$
By the subclass definition:
- $\bigcap A \subseteq S$
By Subclass of Set is Set, $\bigcap A$ is a set.
$\blacksquare$
Sources
- 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Separation Schema