Proposition
232
[1Z9]If we now fix a family \(\mathcal F\) of sets of interest, we first define the relation \(A∼ B\iff | A|= | B|\) in it; it is easily shown that this is an equivalence relation; so we get that \(| A|≤ | B|\) is a total order in \({\mathcal F}/∼\).
Proof
This derives from the Proposition [1Z7], since the relation
\[ ARB \iff | A|≤ | B| \]
is reflexive and transitive, and by Cantor–Bernstein’s Theorem
\[ | A|≤ | B| ∧ | B|≤ | A| \iff A∼ B \quad . \]