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We have the following indirect implication of form equivalence classes:

82 \(\Rightarrow\) 83 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
82 \(\Rightarrow\) 83	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
82:	\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)
83:	\(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite.

Comment: