Implications

We have the following indirect implication of form equivalence classes:

257 \(\Rightarrow\) 324
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
257 \(\Rightarrow\) 260 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
260 \(\Rightarrow\) 40 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
40 \(\Rightarrow\) 165 clear
165 \(\Rightarrow\) 324 clear

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

Howard-Rubin Number Statement
257:

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

260:

\(Z(TR\&C,P)\): If \((X,R)\) is a transitive and connected relation in which every partially ordered subset has an upper bound, then \((X,R)\) has a maximal element.

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

165:

\(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function.

324:

\(KW(WO,WO)\), The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set \(M\) of well orderable sets, there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

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