Choice function and well ordering of all set

Question

Show the following are equivalent:

1. Every set can be well-ordered
2. $\forall S[0\notin S\rightarrow\exists C:S\rightarrow\bigcup S\land(\forall Y\in S(C(Y)\in Y))]$

So I just prove the equivalence between:

1. X can be well-ordered
2. There is a $C:(P(X)\backslash \{0\})\rightarrow X$ such that $\forall Y\subseteq X(Y\neq 0\rightarrow C(Y)\in Y)$

I feel it is the two equivalance are very similar, but I'm not too sure these two can relate to each other and assist the proof of the first one.

Answer 1

Each member of $S$ is a subset of $\bigcup S$, and the empty set is not a member of $S$. So $S$ is a subset of $P(X)\setminus\{0\}$. So you can just restrict your choice function on that to $S$.