I'm trying to understand a particular proof, but it has a step which isn't really given any justification. Here is the part that's giving me trouble:

"Let

be a semiprime ideal of a semigroup

, and let

. Letting

, we obtain an m-system disjoint from

."

For those unfamiliar with the terminology, a semiprime ideal is an ideal of a semigroup

with the property that for any

. An m-system is a non-empty subset

of S such that

there exists

such that

.

What I'm trying to understand is why

should be disjoint from

. I understand why it is an m-system, but I don't understand why, say,

can't be in

. Any insights would be appreciated.