Implications

We have the following indirect implication of form equivalence classes:

29 \(\Rightarrow\) 338
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
29 \(\Rightarrow\) 27 Unions of well-ordered sets, Howard, P. 1994, J. Austral. Math. Soc. Ser. A.
27 \(\Rightarrow\) 31 clear
31 \(\Rightarrow\) 338 clear

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

Howard-Rubin Number Statement
29:  If \(|S| = \aleph_{0}\) and \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets and \(|A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup^{}_{x\in S} A_{x}| = |\bigcup^{}_{x\in S} B_{x}|\). Moore, G. [1982], p 324.

27:

\((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The  union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36.

31:

\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem:  The union of a denumerable set of denumerable sets is denumerable.

338:

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

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