Implications

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We have the following indirect implication of form equivalence classes:

222 \(\Rightarrow\) 142 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
222 \(\Rightarrow\) 142	The strength of the Hahn-Banach theorem, Pincus, D. 1972c, Lecture Notes in Mathematics

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

Howard-Rubin Number	Statement
222:	There is a non-principal measure on \(\cal P(\omega)\).
142:	\(\neg  PB\):  There is a set of reals without the property of Baire.  Jech [1973b], p. 7.

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