Dimension of irreducible projective algebraic set

Question

Let $Y \subset P^n(\Bbb{C}$) is an irreducible projective algebraic set, then how to show that dim$Y$ is equal to the minimum $r \in \Bbb{ N }$ such that there exists a linear subspace $S_{n-r-1} \subset P^n(\Bbb{C}$) with $S_{n-r-1}\cap X=\emptyset$. It will be helpful if i get a reference for this fact.

Answer 1

This is Lemma 3.2 of chapter VII (page 122) of Algebraic Geometry, An Introduction by Perrin.