Corollary 2

Tag 03R5, permanent. Cite it as [Bourbaki, Tag 03R5].

Theory of Sets, chapter III, § 3. Equipotent sets. Cardinals

Let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinals and let $\mathrm{I}$ be a set equipotent to $\mathfrak{b}$ . For each $\iota \in \mathrm{I}$ let $\mathfrak{a}_\iota = \mathfrak{a}$ , $\mathfrak{c}_\iota = 1$ . Then

\mathfrak{a}\mathfrak{b} = \sum_{\iota \in \mathrm{I}} \mathfrak{a}_\iota, \qquad \mathfrak{b} = \sum_{\iota \in \mathrm{I}} \mathfrak{c}_\iota.

The second formula is a consequence of the fact that any set is the union of its one-element subsets. The first formula follows from the second by multiplying by $\mathfrak{a}$ and using Corollary 1.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Proposition 1, § 5
Proposition 9, § 5

Cites

Corollary 1