Implications

We have the following indirect implication of form equivalence classes:

150 \(\Rightarrow\) 358
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
150 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 358 clear

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

Howard-Rubin Number Statement
150:

\(PC(\infty,\aleph_0,\infty)\): Every infinite set of denumerable sets has an infinite subset with a choice function.

32:

\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets  has a choice function.

10:

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

358:

\(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets 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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