Proving that induction works for von Neuman's construction of naturals

Question

I have read on a number of books and other references that the von Neumann ordinals -- the model for ${\mathbb N}$ in which zero is $\emptyset$ and $s(n)=n\cup\{n\}$ -- satisfy all Peano axioms. I can work out the proofs for axioms 1-4, but how would I prove that the induction axiom works on such construction without using induction itself? Is it OK to use induction, since it's metalanguage anyway?