Core-nonemptiness-game theory

Question

In the snippet below in the theorem 3.2.6. there is given a characterization of the core like this: $NE,IR, RGP$. However, I believe that core might be empty so it doesn't satisfy $NE$. What's wrong here with this ?

Answer 1

$NE$ in the theorem $3.2.6$ can be satisfied for all games in question because we are working on $\Gamma$ which is by definition $$\{(N,V)|C(N,V)\neq\emptyset\}.$$