I can't seem to be able to prove Kuratowski's Theorem using the Axiom of Choice, although they are equivalent assertions.
Kuratowski's Lemma: Every partial order has a maximal chain.
Axiom of Choice: For every set X of disjoint nonempty sets there exists a set$Y $such that for every set $Z \in X, Y \cap Z$ is a singleton.
My attempt: Consider any chain $C_0$ of the partial order. If $\exists x \in X \setminus C_0$ which is comparable with some element of $C_0$, let $C_1 := C_0 \cup \{ x \}$. Iterate this process . If at some point we cannot find such an x, then we have found a maximal chain. Suppose we can find such an $x$ infinitely, then the sets $i\geq 1 \Rightarrow X_i := C_{i+1} \setminus C_i$ are disjoint singletons. Hence by axiom of choice there exists $Y$ for which $X_i \subseteq Y$. Inorder to finish the proof, I need to prove something of the form "If a is comparable with some element of $C_0$, then $\exists j$ s.t. $a \in C_j$". I can't seem to prove this.
P.S: x is comparable with y iff $x R y \lor y R x$.