I am trying to solve the following exercise: Let $R$ be a commutative ring (here a ring is assumed to always have a multiplicative identy) and consider the following property for an ideal $I$:
(1) For every set $S$ of prime ideals of $R$, i.e. $S\subseteq spec(R)$, if $I$ is contained in the union of the prime ideals in $S$ it is already contained in one of them.
Show that, if (1) holds for all prime ideals, it already holds for all ideals.
The result reminds me of prime avoidance, but I have no clue how to approach the problem. Any hints would be much appreciated!