I want to prove that $$K_iE\subseteq E,$$ where $K_iE$ denotes the set of facts that are knowledge of agent $i$, and $E$ is the set of all facts.
I have the following definitions:
For an indistinguishability relation, $\sim_i$, we have $\sim_i[w]$ is the set of worlds that agent $i$ cannot distinguish from its current world, which is $\{ w'|(w, w')\in\sim_i \}.$ We also know that $\sim_i[w]\in E$ and $$K_i E =\{w|\sim_i[w]\subseteq E\}.$$
Armed with this, I am struggling to show that $K_iE \subseteq E$. I think maybe an approach would be to let $w=w'$ and use the reflexivity of the indistinguishability relation. So we would have $$\sim_i[w]=\{w|(w, w)\in \sim_i\}.$$ But I am quite unsure. Could someone please help?