Axiom Of Choice

We have the following indirect implication of form equivalence classes:

44 \(\Rightarrow\) 41 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
44 \(\Rightarrow\) 41	The Axiom of Choice, Jech, 1973b, page 120 theorem 8.1

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

Howard-Rubin Number	Statement
44:	\(DC(\aleph _{1})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(\|Y\| < \aleph_{1}\) there is an \(x \in X\) with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\).
41:	\(W_{\aleph _{1}}\): For every cardinal \(m\), \(m \le \aleph_{1}\) or \(\aleph_{1}\le m \).

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