Implications

We have the following indirect implication of form equivalence classes:

60 \(\Rightarrow\) 151
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
60 \(\Rightarrow\) 231 Cardinals and the Boolean prime ideal theorem, Tsukada, N. 1977, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A
231 \(\Rightarrow\) 151 clear

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

Howard-Rubin Number Statement
60:

\(C(\infty,WO)\): Every set of non-empty, well orderable sets has a choice function.
Moore, G. [1982], p 125.

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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