About prime ideals in lattice theory

Question

Can anyone help me with this question? If I and J are two ideals in a complete lattice and their intersection is contained in a prime ideal P ,then is it true that either I is a subset of P or J is a subset of P

Answer 1

Suppose that $I \cap J \subseteq P$ and $I \nsubseteq P$, and conclude that $J \subseteq P$.

If $I \nsubseteq P$ then there exists $x \in I$ such that $x \notin P$, whence $x \notin J$.
Take $y \in J$ (we wish to prove that $y \in P$).
Since $x \in I$ and $y \in J$, we have $x \wedge y \in I \cap J$. (Can you see why?)
Thus $x \wedge y \in P$. As $P$ is prime and $x \notin P$, we must have $y \in P$.