A fast 3D collision detection algorithm

I'm trying to work through the math here, and I don't understand why these two propositions are equivalent:

1) min_{x,y} |x-y|^2

   x ∈ A

   y ∈ B

2) = min_{x,y} d

   d ≥ |x-y|^2

   x ∈ A

   y ∈ B

What is 'd'? If d is much greater than |x-y|^2 at the actual (x, y) with minimal distance, and equal to |x-y|^2 at some other (x', y'), couldn't (2) yield a different, wrong solution? Is it implied that 'd' is a measure or something, such that it's somehow constrained or bounded to prevent this?