Set of Natural Numbers Equals Union of its Successor

Theorem
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction on a Zermelo universe $V$.

Then:
 * $\bigcup \omega^+ = \omega$

Proof
We have that:
 * $\omega^+ = \omega \cup \set \omega$

and so:
 * $\omega \subseteq \omega^+$

By definition:
 * $\bigcup \omega^+ = \set {x: \exists y \in \omega^+: x \in y}$

Thus:
 * $x \in \bigcup \omega^+ \implies x \in \omega$

So by definition of set equality:


 * $\bigcup \omega^+ = \omega$