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We have the following indirect implication of form equivalence classes:

221 \(\Rightarrow\) 222 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
221 \(\Rightarrow\) 222	clear

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

Howard-Rubin Number	Statement
221:	For all infinite \(X\), there is a non-principal measure on \(\cal P(X)\).
222:	There is a non-principal measure on \(\cal P(\omega)\).

Comment: