Axiom of Choice implies Hausdorff's Maximal Principle/Proof 1
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Theorem
Let the Axiom of Choice be accepted.
Then Hausdorff's Maximal Principle holds.
Proof
Let $S$ be the set of all chains of $\PP$.
$S \ne \O$ since the empty set is an element of $S$.
From Subset Relation is Ordering, we have that $\struct {S, \subseteq}$ is partially ordered by inclusion.
Let $C$ be a totally ordered subset of $\struct {S, \subseteq}$.
Then $\bigcup C$ is a chain in $C$ by Set of Chains is Closed under Chain Unions under Subset Relation.
This shows that $S$, ordered by set inclusion, is an inductive ordered subset.
By applying Zorn's Lemma, the result follows.
$\blacksquare$
Source of Name
This entry was named for Felix Hausdorff.