Implications

We have the following indirect implication of form equivalence classes:

405 \(\Rightarrow\) 390
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
405 \(\Rightarrow\) 75 clear
75 \(\Rightarrow\) 404 clear
404 \(\Rightarrow\) 390 clear

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

Howard-Rubin Number Statement
405:

Every infinite set can be partitioned into sets each of which is countable and has at least two elements.

75:

If a set has at least two elements, then it can be partitioned into well orderable subsets, each of which has at least two elements.

404:

Every infinite set can be partitioned into infinitely many sets, each of which has at least two elements. Ash [1983].

390:

Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983].

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