Let $A$ be a commutative ring with unit and $\mathcal{M}$ the set of all the maximal ideals of $A$.
I need to prove that $$\mathfrak{m}\not \subseteq \bigcup_{\mathfrak{a}\in \mathcal{M}\setminus \{\mathfrak{m}\}} \mathfrak{a}$$ for every $\mathfrak{m}\in \mathcal{M}$.
I get to this statement trying to prove a remark on Neukirch's Algebraic Number Theory that says that if $\mathcal{F}$ is a set of prime ideals of $A$ omitting only finitely many primes, then if $S=(\cup_{\mathfrak{p}\in \mathcal{F}}\mathfrak{p})^c$ we have that the prime ideals of $AS^{-1}$ are of the form $\mathfrak{q}S^{-1}$ where $\mathfrak{q}$ is a prime ideal of $A$ contained in a prime ideal of $\mathcal{F}$ [Page 66]. (I was able to prove that the statement above is equivalent to this)
