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.

$\blacksquare$