Iwamura’s Lemma, Markowsky’s Theorem, and ordinals

On p.61 of the book, there is a remark that the dcpos are exactly the chain-complete posets.  This is a theorem by George Markowsky (1976).  It is time I explained seriously how this worked.  The first step is Iwamura’s Lemma (1944), which states that every directed subset decomposes as the union of a small chain of small directed subsets.

The reason I did not put the proof of that result in the book is because it rests on using ordinals, and I did not want to introduce ordinals, specially if they served for only one result.  I’m attacking the problem head-front in the full post.