Element of Class is Subset of Union of Class
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Theorem
Let $A$ be a class.
Let $\ds \bigcup A$ denote the union of $A$.
Let $x \in A$.
Then:
- $x \subseteq \ds \bigcup A$
Proof
Let $x \in A$.
By definition of class, $x$ is a set.
Let $y \in x$.
By definition of union of $A$:
- $\ds \bigcup A := \set {y: \exists x \in A: y \in x}$
It follows directly from that definition that:
- $y \in \ds \bigcup A$
The result follows by definition of subset.
$\blacksquare$