Implications

We have the following indirect implication of form equivalence classes:

265 \(\Rightarrow\) 263
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
265 \(\Rightarrow\) 143 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
143 \(\Rightarrow\) 263 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
265:

\(H(A,TR)\): Every relation \((X,R)\) contains a \(\subseteq\)-maximal transitive subset.

143:

\(H(C,TR)\): If \((X,R)\) is a connected relation (\(u\neq v\rightarrow u\mathrel R v\) or \(v\mathrel R u\)) then \(X\) contains a \(\subseteq\)-maximal transitive subset.

263:

\(H(AS\&C,P)\): Every every relation \((X,R)\) which is antisymmetric and connected contains a \(\subseteq\)-maximal partially ordered subset.

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