Prime of codimension n is minimal over an ideal generated by n elements

Here the ring R is assumed to be Noetherian. Suppose P is a prime of codimensin n.
1. Every minimal prime(over (0)) has codimension 0.
2. Every prime contains a minimal prime. If x1 is in P but not in any of the minimal primes, then the primes minimal over (x1) strictly contains a minimal prime, and hence has codimension 1, by Principal Ideal Theorem.
2. If x2 is in P but not in any of the minimal primes over (x1), then the primes minimal over (x1, x2) strictly contains a minimal prime over (x1), and hence has codimension 2 by Principal Ideal Theorem.
3. This process can't continue past k=n, since it would imply P contains a prime of codimension greater than n. The process can't stop before k=n either, because that would imply that P is contained in the unions of primes of codimension less than n, which implies that P is equal to one of them.
4. So this process will stop when k=n, and this implies that P is minimal over (x1, ..., xn). QED