Proposition 5

The set $\mathfrak{F}(\mathrm{E})$ of finite subsets of an infinite set E is equipotent to E.

For each integer $n$, let $\mathfrak{F}_n$ denote the set of subsets of E which have $n$ elements. For each $\mathrm{X} \in \mathfrak{F}_n$ there exists a bijection of $[1, n]$ onto X. Hence the cardinal of $\mathfrak{F}_n$ is at most equal to that of the set of mappings of $[1, n]$ into E, i.e., to $\mathrm{Card}(\mathrm{E}^n) = \mathrm{Card}(\mathrm{E})$ (no. 3, Theorem 2, Corollary 1). Therefore

On the other hand, since $x \to \{x\}$ is an injective mapping of E into $\mathfrak{F}(\mathrm{E})$, we have $\mathrm{Card}(\mathrm{E}) \leqslant \mathrm{Card}(\mathfrak{F}(\mathrm{E}))$.