Implications

We have the following indirect implication of form equivalence classes:

376 \(\Rightarrow\) 167
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
376 \(\Rightarrow\) 167 clear

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

Howard-Rubin Number Statement
376:

Restricted Kinna Wagner Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) and a function \(f\) such that for every \(z\subseteq Y\), if \(|z| \ge 2\) then \(f(z)\) is a non-empty proper subset of \(z\).

167:

\(PKW(\aleph_{0},\ge 2,\infty)\), Partial Kinna-Wagner Principle:  For every denumerable family \(F\) such that for all \(x\in F\), \(|x|\ge 2\), there is an infinite subset \(H\subseteq F\) and a function \(f\) such that for all \(x\in H\), \(\emptyset\neq f(x) \subsetneq x\).

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