Corollary

Tag 04TW, permanent. Cite it as [Bourbaki, Tag 04TW].

Commutative Algebra, chapter VI, § 4. The height of a valuation

The set & subrings of K containing A is totally ordered by inclusion.

The set of prime ideals of $A$ is totally ordered by inclusion ( $\S$ 1, no. 2, Theorem 1 (e)) and the mapping $\mathfrak{p} \mapsto A_{\mathfrak{p}}$ reverses the inclusion relations.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 4. The height of a valuation
Proposition 4, § 7

Cites

§$ 1, no. 2, Theorem 1