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We have the following indirect implication of form equivalence classes:

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

Implication	Reference
257 \(\Rightarrow\) 0

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

Howard-Rubin Number	Statement
257:	\(Z(TR,P)\): Every transitive relation \((X,R)\) in which  every partially ordered subset has an upper bound, has a maximal element.
0:	\(0 = 0\).

Comment: