Kunen, II.56. Having trouble proving the properties of the following:
The definition: $S(\kappa,\lambda,\mathbb{I})$ is the statement that $\kappa > \omega$ and $\mathbb{I}$ is a $\kappa$-complete ideal on $\kappa$ which contains each singleton and which is $\lambda$-saturated, meaning: there is no family $\{X_\alpha : \alpha < \lambda\}$, such that each $X_\alpha \notin \mathbb{I}$ but $\alpha \ne \beta \rightarrow (X_\alpha \cap X_\beta) \in \mathbb{I}$. Need to show that:
a) $\exists\lambda\exists\mathbb{I} S(\kappa,\lambda,\mathbb{I}) \rightarrow \kappa$ is regular.
b) $S(\kappa,\lambda,\mathbb{I}) \land \lambda < \lambda' \rightarrow S(\kappa,\lambda',\mathbb{I})$
c) $\exists\mathbb{I}S(\kappa,\kappa,\mathbb{I}) \rightarrow \kappa$ is weakly inaccesible.
I think, the main problem for me here, is not knowing what sets are in or out of $\mathbb{I}$.
I'll appreciate any help. Thanks in advance.