Prime Avoidance Lemma
Theorem. Suppose $p_1,\dots,p_n$ are prime ideals. Take an ideal $I$ such that $I\subset \bigcup_i p_i$. Then $I$ is contained entirely in some prime ideal: $I\subset p_i$ for some $i$.
Theorem. Suppose $p_1,\dots,p_n$ are prime ideals. Take an ideal $I$ such that $I\subset \bigcup_i p_i$. Then $I$ is contained entirely in some prime ideal: $I\subset p_i$ for some $i$.