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. 


Question: Does "so far, so good" carry a negative connotation?
Answer: "So far so good" means that things are "fine" in the up-to-now sense of the word. What your wife found missing in the phrase was an "all clear" for the future.