Proposition 9

Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a family of sets. Then there exists a set X with the following property : X is the union of a family $(\mathrm{X}'_\iota)_{\iota \in \mathrm{I}}$ of mutually disjoint sets such that for each $\iota \in \mathrm{I}$ there exists a one-to-one mapping of $\mathrm{X}_\iota$ onto $\mathrm{X}'_\iota$.

Let $\mathrm{A} = \bigcup_{\iota \in \mathrm{I}} \mathrm{X}_\iota$. If $\iota \in \mathrm{I}$, the mapping $x \rightarrow (x, \iota)$ $(x \in \mathrm{X}_\iota)$ is a one-to-one mapping of $\mathrm{X}_\iota$ onto a subset $\mathrm{X}'_\iota$ of $\mathrm{A} \times \mathrm{I}$. Moreover, the image of $\mathrm{X}'_\iota$ under the second coordinate function on $\mathrm{A} \times \mathrm{I}$ is contained in the set $\{\iota\}$; it follows that $\mathrm{X}'_\iota \cap \mathrm{X}'_\varkappa = \emptyset$ whenever $\iota \neq \varkappa$. We may then take $\mathrm{X} = \bigcup_{\iota \in \mathrm{I}} \mathrm{X}'_\iota$.