Show that for all $\textbf{n} \in \mathbb{N}$, if $x \in \textbf{n}$ then $x \in \mathbb{N}$. For reference, this question is taken from page 47 of 'Classic Set Theory' by Derek Goldrei.
The chapter is called 'the construction of the natural numbers', so not everything we 'typically' know about natural numbers is known at this point. We have so far that: $\mathbb{N}$ with element $0$ (the empty set) and the successor function $S : \textbf{n} \longrightarrow \textbf{n}^+$ satisfies Peano's axioms (http://mathworld.wolfram.com/PeanosAxioms.html). 
We know that $\mathbb{N}$ is a well-ordered set, well ordered by $\in$ but we do not know the usual properties for addition, multiplication and exponentiation hold (associativity, commutativity etc). 
Any help would be greatly appreciated. Thanks. 

HINT:
Assuming that you define $n^+$ as the set $n\cup\{n\}$, then by induction you can show that if $n$ is a natural number then all its elements are natural numbers.
