Inductive Construction of Natural Numbers fulfils Peano's Axiom of Injectivity

Theorem
Let $P$ denote the set of natural numbers by definition as an inductive set.

Then $P$ fulfils:

where $s$ denotes the successor mapping.

Proof
Let $m$ and $n$ be natural numbers such that $n^+ = m^+$.

By construction:
 * $n \in n^+$

and:
 * $m \in m^+$

Thus as $n^+ = m^+$ we have:
 * $n \in m^+$

and:
 * $m \in n^+$

This gives us:
 * $n \in m \lor n = m$

and:
 * $m \in n \lor m = n$

that $n \ne m$.

Then from $n \in m \lor n = m$ we have:
 * $n \in m$

and from $m \in n \lor m = n$ we have:
 * $m \in n$

In summary, if $n \ne m$ we have
 * $n \in m$ and $m \in n$

But from Natural Numbers cannot be Elements of Each Other, this is not possible.

Hence by Proof by Contradiction:
 * $n^+ = m^+ \implies n = m$

and the result follows by definition of injection.