Element of Class is Subset of Union of Class

Theorem
Let $A$ be a class.

Let $\displaystyle \bigcup A$ denote the union of $A$.

Let $x \in A$.

Then:
 * $x \subseteq \displaystyle \bigcup A$

Proof
Let $x \in A$.

By definition of class, $x$ is a set.

Let $y \in x$.

By definition of union of $A$:


 * $\displaystyle \bigcup A := \set {y: \exists x \in A: y \in x}$

It follows directly from that definition that:
 * $y \in \displaystyle \bigcup A$

The result follows by definition of subset.