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Order the collection $\mathrm{Ch}(P)$ of all chains in a partial order, including the empty chain, by inclusion. Suppose that $f:\mathrm{Ch}(P)\to P$ were an order-preserving injection. The chain-poset nonembedding lemma follows by a transfinite recursion of length the Hartogs ordinal $h(P)$:Inductively, the previously constructed points form a strictly increasing chain. If $\beta <\alpha$, then $C_{\beta }⊊C_{\alpha }$, since $p_{\beta }$ belongs to the latter but not the former. Monotonicity gives $p_{\beta }\le p_{\alpha }$, and injectivity makes this inequality strict. Thus the recursion continues at every $\alpha <h(P)$ and gives an injection $h(P)\to P$, contradicting the defining property of the Hartogs ordinal. Therefore $\boxed{\mathrm{Ch}(P)\text{ has no order-preserving embedding into }P.}$ The empty chain is part of the convention: excluding it would make a one-point poset a counterexample.

Order the collection $\mathrm{Ch}(P)$ of every chain in a partial order, including the empty chain, by inclusion. Suppose that $j:\mathrm{Ch}(P)\to P$ is an order-preserving function and an injection. Let $\theta =h(P)$ be the Hartogs ordinal: by Hartogs theorem, no injection from $\theta$ to $P$ exists.

Use transfinite recursion to define, for $\alpha <\theta$,We verify both that $j(C_{\alpha })$ is defined and that the values are strictly increasing. Assume inductively that the earlier values are strictly increasing. Then $C_{\alpha }$ is a chain in a partial order. For any $\beta <\alpha$, $C_{\beta }\subseteq C_{\alpha }$, so the fact that $j$ is order-preserving gives $p_{\beta }\le p_{\alpha }$. Furthermore $p_{\beta }\in C_{\alpha }$, whereas $p_{\beta }\notin C_{\beta }$ by distinctness of the earlier values. Thus $C_{\beta }\ne C_{\alpha }$, and injectivity gives $p_{\beta }\ne p_{\alpha }$. We have provedThe empty initial chain starts the induction, and the same argument applies at every limit ordinal.

If a globally defined recursion rule is desired, apply $j$ when the earlier range is a chain and otherwise use the fixed value $j(\varnothing )$. The induction shows that the fallback case never occurs. Thus no choices of new points are being made: the function $j$ uniquely determines every stage.

The resulting function $\alpha \mapsto p_{\alpha }$ is an injection from $\theta$ to $P$, contrary to its defining Hartogs ordinal property. Hence the chain-poset nonembedding lemma gives