Non-Empty Bounded Subset of Natural Numbers has Greatest Element

Theorem
Let $\omega$ be the set of natural numbers defined as the von Neumann construction.

Then every non-empty bounded subset of $\omega$ has a greatest element.

Proof
From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.

From Successor Mapping on Natural Numbers is Progressing, this successor mapping is a progressing mapping.

The result is a direct application of Non-Empty Bounded Subset of Minimally Inductive Class under Progressing Mapping has Greatest Element.