Prove that every ideal in $\Bbb{Z}$ is a principal ideal.

Question

I have a question about the following proof for the theorem:

Prove that every ideal in $\Bbb{Z}$ is a principal ideal.

In Case 2, I do not understand how the principle of well-ordering is used. How it is known that $I$ is a subset of $\Bbb{N}$?

Answer 1

Note that the set $\lbrace z \in I\;|\; z \in \mathbb{N} \rbrace \neq \emptyset$. For this, take $0, a \in I$ so $$0 - a\;\mathrm{and}\;0+a \in I$$ Thus, $a, -a \in I$. Therefore, theres exists a subset of natural numbers contained in $I$.