Chain-poset nonembedding lemma ($\mathrm{Ch}(P)/{\hookrightarrow }_{\mathrm{monotone}}P$)

If $\mathrm{Ch}(P)$ 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 $P$.