Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
421 \(\Rightarrow\) 338 clear
338 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 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
421:   \(UT(\aleph_0,WO,WO)\): The union of a denumerable set of well orderable sets can be well ordered.
338:

\(UT(\aleph_0,\aleph_0,WO)\): The union of a denumerable number of denumerable sets is well orderable.

32:

\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable 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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