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Proof

(related to Proposition: Cardinal Number)

It is sufficient to show that being equipotent is an equivalence relation $"\sim"$ on the sets belonging to a given (non-empty) set system $\mathcal X$. In the following, let $A,B,C\subseteq\mathcal X.$

• Since there is always a bijective function $f:A\to A$ (i.e. the set $A$ is equipotent to itself), we have that $A\sim A$. Therefore, "$\sim$" is reflexive.
• Let $A\sim B$. Since there is bijective function $f:A\to B$, it is invertible with $f^{-1}:B\to A.$ Thus, $B\sim A$ and "$\sim"" is symmetric.
• Let $A\sim B$ and $B\sim C$. Therefore there are bijective maps $f_1:A\to B$ and $f_2:B\to C$. Since the composition of bijective functions is bijective, the is bijective function $f:=f_2\circ f_1 :A\to C$. Thus $A\sim C$, and "$\sim"" is transitive.
• Therefore, "$\sim$" is an equivalence relation, by definition.
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References

Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition