Axiom Of Choice

We have the following indirect implication of form equivalence classes:

59-le \(\Rightarrow\) 105 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
59-le \(\Rightarrow\) 105	clear

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

Howard-Rubin Number	Statement
59-le:	If \((A,\le)\) is a partial ordering that is not a well ordering, then there is no set \(B\) such that \((B,\le)\) (the usual injective cardinal ordering on \(B\)) is isomorphic to \((A,\le)\). Mathias [1979], p 120.
105:	There is a partially ordered set \((A,\le)\) such that for no set \(B\) is \((B,\le)\) (the ordering on \(B\) is the usual injective cardinal ordering) isomorphic to \((A,\le)\).

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