Implications

We have the following indirect implication of form equivalence classes:

15 \(\Rightarrow\) 404
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
15 \(\Rightarrow\) 296 The dense linear ordering principle, Pincus, D. 1997, J. Symbolic Logic
296 \(\Rightarrow\) 404 clear

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

Howard-Rubin Number Statement
15:

\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every  set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).  

296:

Part-\(\infty\): Every infinite set is the disjoint union of infinitely many infinite sets.

404:

Every infinite set can be partitioned into infinitely many sets, each of which has at least two elements. Ash [1983].

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