Set of Natural Numbers Equals Union of its Successor

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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:

$\bigcup \omega^+ = \omega$

$\blacksquare$


Sources