The von Neumann naturals form a Peano system

(P3) The induction axiom holds: it is exactly the induction principle.

(P1) For every $n$, $\sigma(n) = n \cup \{n\}$ contains $n$, so $\sigma(n)$ is nonempty and $\sigma(n) \neq \varnothing = 0$.

(P2) Suppose $\sigma(m) = \sigma(n)$, i.e. $m \cup \{m\} = n \cup \{n\}$; then $m \in n \cup \{n\}$ so $m \in n$ or $m = n$, and symmetrically $n \in m$ or $n = m$; if $m \neq n$ then $m \in n$ and $n \in m$, whence by transitivity [L2] $m \in n$ gives $m \subseteq n$, and $n \in m \subseteq n$ gives $n \in n$, contradicting $n \notin n$ [L2]; therefore $m = n$ and $\sigma$ is injective.

P1, P2, P3 all hold, so $(\mathbb{N}, 0, \sigma)$ is a Peano system; in particular a Peano system exists.