Prime ideal need not contain any prime element

Statement
It is possible to have an integral domain $$R$$ and a nonzero fact about::prime ideal $$P$$ of $$R$$ such that $$P$$ does not contain any fact about::prime element of $$R$$.

For unique factorization domains instead of arbitrary integral domains (strengthening of hypothesis)

 * Unique factorization implies every nonzero prime ideal contains a prime element
 * Unique factorization and one-dimensional iff principal ideal
 * Unique factorization and finite-dimensional implies every prime ideal is generated by a set of primes of size at most the dimension
 * Unique factorization and Noetherian implies every prime ideal is generated by finitely many prime elements

For irreducible elements instead of prime elements (weakening of conclusion)

 * ACCP implies every nonzero prime ideal contains an irreducible element
 * Noetherian domain implies every prime ideal is generated by finitely many irreducible elements

Proof
For an example, we can take any fact about::Dedekind domain that is not a principal ideal domain, and pick a prime ideal in the Dedekind domain that is not principal. A concrete example is:

$$R = \mathbb{Z}[\sqrt{-5}], P = (2, 1 + \sqrt{-5})$$.