Corollary 1

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

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

Let $\mathfrak{a}$ be a cardinal and let $(\mathfrak{b}_{\iota})_{\iota \in I}$ be a family of cardinals. Then

\mathfrak{a}^{\sum_{\iota \in I} \mathfrak{b}_{\iota}} = \mathbf{P}_{\iota \in I} \mathfrak{a}^{\mathfrak{b}_{\iota}}.

Let S be the sum of the sets $\mathfrak{b}_{\iota}$ , and put $\mathfrak{a}_{s} = \mathfrak{a}$ for all $s \in S$ . Both sides of the equality to be proved are then equal to $\mathbf{P}_{s \in S} \mathfrak{a}_{s}$ , by virtue of Proposition 10 and the associativity formula for products (no. 3, Proposition 5(b)).

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

Corollary 3, § 3

Cites

Proposition 10
no. 3, Proposition 5