Is it true that $0\in 1$?

Yes, $0\in1$, since $0=\emptyset$ and $1=\{\emptyset\}$. On the other hand, you are wrong when you assert that Peano axioms assert that $0=\emptyset$ and $1=\{\emptyset\}$.

It goes like this:

• $0=\emptyset $
• $n+1=n \cup\{n\}$

So $n=\{0,1,...,n-1\}$.

So $$m \leq n \implies m \subseteq n$$ $$m<n \implies m \in n$$