Axiom Of Choice

We have the following indirect implication of form equivalence classes:

379 \(\Rightarrow\) 379 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
379 \(\Rightarrow\) 379

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

Howard-Rubin Number	Statement
379:	\(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\).
379:	\(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\).

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