= Chain-poset nonembedding lemma
{title2=$\operatorname{Ch}(P)\not\hookrightarrow_{\rm monotone}P$}
If $\operatorname{Ch}(P)$ is the inclusion-ordered collection of every <chain in a partial order>, including the empty chain, it has no <order-preserving> injection into $P$. Supposing $j$ exists, recursively set $p_\alpha=j(\{p_\beta:\beta<\alpha\})$ for $\alpha<h(P)$. By induction the earlier values form a chain; its proper initial chains are distinct, so monotonicity and injectivity make $p_\alpha$ strictly greater than every earlier value. This would inject the <Hartogs ordinal> into $P$.
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