Implications

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 151
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 \(\Rightarrow\) 40 clear
40 \(\Rightarrow\) 231 Abzählbarkeit und Wohlordenbarkeit, Felgner, U. 1974, Comment. Math. Helv.
231 \(\Rightarrow\) 151 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.

40:

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

231:

\(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable.

151:

\(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well  orderable. (If \(\kappa\) is a well ordered cardinal, see note 27 for \(UT(WO,\kappa,WO)\).)

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