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We have the following indirect implication of form equivalence classes:

84 \(\Rightarrow\) 84 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
84 \(\Rightarrow\) 84	Definitions of finite, Howard, P. 1989, Fund. Math.

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

Howard-Rubin Number	Statement
84:	\(E(II,III)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite).
84:	\(E(II,III)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite).

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