Implications

We have the following indirect implication of form equivalence classes:

302 \(\Rightarrow\) 249
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
302 \(\Rightarrow\) 392 clear
392 \(\Rightarrow\) 393 clear
393 \(\Rightarrow\) 121 clear
121 \(\Rightarrow\) 122 clear
122 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 249

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

Howard-Rubin Number Statement
302:

Any continuous surjection between compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain.

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.

249:

If \(T\) is an infinite tree in which every element has exactly 2 immediate successors then \(T\) has an infinite branch.

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