Implications

We have the following indirect implication of form equivalence classes:

20 \(\Rightarrow\) 26
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
20 \(\Rightarrow\) 22
22 \(\Rightarrow\) 26 clear

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

Howard-Rubin Number Statement
20:

If \(\{A_{x}: x \in S \}\) 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 [1982] (1.4.12 and 1.7.8).

22:

\(UT(2^{\aleph_{0}},2^{\aleph_{0}},2^{\aleph_{0}})\): If every member of an infinite  set of cardinality \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\), then the union has power \(2^{\aleph_{0}}\).

26:

\(UT(\aleph_{0},2^{\aleph_{0}},2^{\aleph_{0}})\): The union of denumerably many sets each of power \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\).

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