# 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:

$\omega \subseteq \omega^+$

$\Box$

By definition:

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

Thus:

$x \in \bigcup \omega^+ \implies x \in \omega$

$\Box$

So by definition of set equality:

$\bigcup \omega^+ = \omega$

$\blacksquare$