Lemma 15.4 of Jech: Let $\kappa$ be a regular cardinal such that $2^{< \kappa} = \kappa$. Let $S$ be an arbitrary set and let $|C| \leq \kappa$. Let $P$ be the set of all functions $p$ whose domains are subsets of $S$ of size $< \kappa$ with values in $C$. Let $p < q$ iff $p \supset q$. Then $P$ satisfies the $\kappa^+$-chain condition.
Proof: Let $W \subset P$ be an antichain. We construct sequences $A_0 \subset A_1 \subset ... \subset A_{\alpha} \subset ... (\alpha < \kappa)$ of subsets of $S$ and $W_0 \subset ... \subset W_{\alpha} \subset ... (\alpha < \kappa)$ of subsets of $W$. If $\alpha$ is a limit ordinal, let $W_{\alpha} = \bigcup_{\beta < \alpha} W_{\beta}$ and $A_{\alpha} = \bigcup_{\beta < \alpha} A_{\beta}$. Given $A_{\alpha}$ and $W_{\alpha}$, we choose for each $p \in P$ with dom$(p) \subset A_{\alpha}$ some $q \in W$ (if there is one) such that $p = q|_{A_{\alpha}}$. Then we let $W_{\alpha+1} = W_{\alpha} \cup \{\text{the chosen $q$'s}\}$ and $A_{\alpha+1} = \bigcup \{\text{dom}(q) : q \in W_{\alpha+1}\}$. Finally, $A = \bigcup_{\alpha < \kappa} A_{\alpha}$. (The proof continues)
My problem: Who is $A_0$? How do I guarantee that $A_1$ and $W_1$ are not empty?