A theorem of Sierpiński on triads and the axiom of choice (Q2396026)

Sierpiński has proven the following Theorem S: If \(E\) is an infinite set then there is a family \(F\) of three-element subsets of \(E\) such that each two-element subset of \(E\) is included in exactly one member of \(F\). The author shows that Theorem S implies the axiom of choice, as follows: Given an infinite set \(M\), consider \(E = M\cup \aleph(M)\) (Hartogs' function). Take \(F\) as asserted in Theorem S. Let \(P\) denote the set of all two-element subsets of \(\aleph(M)\). For \(m\in M\), let \[ P_m = \{x\in P; \{m\} \cup x\in F\}. \] Then \(P_m\ne 0\), all \(m\in M\) [the assumption of the contrary leads to a one-one map from \(\aleph(M)\) into \(M\)]. Furthermore, \(P_m\cap P_n= 0\) if \(m\ne n\). Since \(P\) can be well-ordered, there is a one-one map from \(M\) into \(P\): \(f(m) = \text{ least element of }P_m\). Therefore, \(M\) can be well-ordered.