Implications

We have the following indirect implication of form equivalence classes:

1 \(\Rightarrow\) 137-k
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
1 \(\Rightarrow\) 137-k

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

Howard-Rubin Number Statement
1:

\(C(\infty,\infty)\):  The Axiom of Choice: Every  set  of  non-empty sets has a choice function.

137-k:

Suppose \(k\in\omega-\{0\}\). If \(f\) is a 1-1 map from \(k\times X\) into \(k\times Y\) then there are partitions \(X = \bigcup_{i \le k} X_{i} \) and \(Y = \bigcup_{i \le k} Y_{i} \) of \(X\) and \(Y\) such that \(f\) maps \(\bigcup_{i \le k} (\{i\} \times  X_{i})\) onto \(\bigcup_{i \le k} (\{i\} \times  Y_{i})\).

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