Krull's height theorem

Statement

Let $R$ be a Noetherian commutative unital ring and $x_1,x_2,\dots ,x_c$ be elements in $R$. Let $P$ be minimal among primes containing all the $x_i$s. Then, the codimension of $P$ is at most $c$.

There is also a converseof this statement viz converse of Krull's height theorem.