Successor Mapping on Natural Numbers has no Fixed Element

Theorem
Let $\N$ denote the set of natural numbers.

Then:
 * $\forall n \in \N: n + 1 \ne n$

Proof
Consider the set of natural numbers as defined by the von Neumann construction.

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

Let $s: \N \to \N$ denote this successor mapping:
 * $\forall x \in \N: \map s x := x + 1$

$\exists n \in \N: n = n + 1$

From Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element, $n$ is the greatest element of $\N$.

From Minimally Inductive Class under Progressing Mapping with Fixed Element is Finite it follows that $\N$ is a finite set.

This contradicts the fact that the natural numbers are by definition countably infinite.