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We have the following indirect implication of form equivalence classes:

31 \(\Rightarrow\) 0 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
31 \(\Rightarrow\) 0

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

Howard-Rubin Number	Statement
31:	\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem:  The union of a denumerable set of denumerable sets is denumerable.
0:	\(0 = 0\).

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