Axiom Of Choice

We have the following indirect implication of form equivalence classes:

261 \(\Rightarrow\) 259 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
261 \(\Rightarrow\) 256	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
256 \(\Rightarrow\) 259	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
261:	\(Z(TR,T)\): Every transitive relation \((X,R)\) in which every subset which is a tree has an upper bound, has a maximal element.
256:	\(Z(P,F)\): Every partially ordered set \((X,R)\) in which every forest \(A\) has an upper bound, has a maximal element.
259:	\(Z(TR\&C,W)\): If \((X,R)\) is a transitive and connected relation in which every well ordered subset has an upper bound, then \((X,R)\) has a maximal element.

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