Implications

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 398
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
264 \(\Rightarrow\) 202 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
202 \(\Rightarrow\) 398 clear

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

Howard-Rubin Number Statement
264:

\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.

202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

398:

\(KW(LO,\infty)\), The Kinna-Wagner Selection Principle for a linearly ordered family of sets: For every linearly ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).

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