Implications

We have the following indirect implication of form equivalence classes:

261 \(\Rightarrow\) 257
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
261 \(\Rightarrow\) 262 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
262 \(\Rightarrow\) 257 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.

262:

\(Z(TR,R)\): Every transitive relation \((X,R)\) in which every ramified subset \(A\) has an upper bound, has a maximal element.

257:

\(Z(TR,P)\): Every transitive relation \((X,R)\) in which  every partially ordered subset has an upper bound, has a maximal element.

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