Implications

We have the following indirect implication of form equivalence classes:

391 \(\Rightarrow\) 18
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
391 \(\Rightarrow\) 392 clear
392 \(\Rightarrow\) 393 clear
393 \(\Rightarrow\) 121 clear
121 \(\Rightarrow\) 122 clear
122 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 18 clear

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

Howard-Rubin Number Statement
391:

\(C(\infty,LO)\): Every set of non-empty linearly orderable sets has a choice function.

392:

\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.

393:

\(C(LO,WO)\): Every linearly ordered set of non-empty well orderable sets has a choice function.

121:

\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.

122:

\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.

10:

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

18:

\(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

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