Section 5.2-5.5, part 16 Proof of the Comparability Theorem

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Im a little confused. It seems to me that, when using zorns lemma, you only proved that every linearly ordered subset of X has an upper bound in the union of all sets which is in relation to your subset. This upper bound would thus change with the choice of the collection of sets. To my understanding, you have to find a set that is the upper bound with respect to all thinkable linear subsets of X. Or did you show this?

adrianott