Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
385 \(\Rightarrow\) 70 note-150
70 \(\Rightarrow\) 222 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
385:

Countable Ultrafilter Theorem:  Every proper filter with a countable base over a set \(S\) (in \({\cal P}(S)\)) can be extended to an ultrafilter.

70:

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

222:

There is a non-principal measure on \(\cal P(\omega)\).

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