Corollary 3
Tag 049T, permanent. Cite it as [Bourbaki, Tag 049T].
Commutative Algebra, chapter II, § 4. Spectra of rings and supports of modules
Let (f_λ)_{λ ∈ L} be a family of elements of A. For an element g ∈ A to satisfy X_g ⊂ ∪_{λ ∈ L} X_{f_λ}, it is necessary and sufficient that there exist an integer n > 0 such that g^n belongs to the ideal generated by the f_λ.
The relation X_g ⊂ ∪_{λ ∈ L} X_{f_λ} is equivalent to V(g) ⊃ ∩_{λ ∈ L} V(f_λ) and it is sufficient to apply Corollary 2.
COROLLARY 4, For two elements f, g of A to satisfy X_f = X_g, it is necessary and sufficient that there exist two integers m > 0, n > 0 such that f^m ∈ Ag and g^n ∈ Af.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.