I have to prove that given any poset $(P,\preceq)$ there exists a chain $S$ such that it is maximal (meaning that if S\subseteq S' then $S=S'$). The book contains a proof using the axiom of choice. The homework assignment is asking me to give an alternate proof using the well-ordering theorem (which is actually equivalent to choice) and transfinite recursion theorem. I managed to prove it in a way that I will show below, however I do not know how to use the transfinite recursion theorem (I might be using it without even knowing it), so if someone can help me point out with I need the transfinite recursion theorem I will be very grateful.
Proof: Assume that $(P,\preceq)$ is a poset. By the Well-ordering theorem, we know that we can impose a well ordering in any set, let this order by given $\leq$ (i.e., $(P,\leq)$ is well ordered). Consider the definite unary condition $H$: $H(x) = \left\{ \begin{array}{ll} \{x\} & \mbox{if }x\in P \mbox{ and }x\cup\bigcup_{y