Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following:
"Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ implies that $I\subset p_j$ for some $j\in \{1, \dots, n\}.$ "
This is clearly equivalent to saying that "if $I$ is not a subset of $p_i$ for any $i,$ then $I$ is not a subset of $\cup_i p_i.$"
But they then claim that this last statement is logically equivalent to the following:
"For each $i,$ there exists $x_i \in I$ such that $x_i \notin p_k$ for all $k\neq i.$"
Can anyone help me to see why this statement is logically a consequence of the first? I know this is elementary, but I'm truly stuck and I'm hoping that someone can phrase things in a way that makes them more transparent.