Russel's paradox: what is the contradiction with $R \not\in R$?

Question

Let the Russel's Set be:

$$R = \{S | S \notin S\}$$ Where $S$ is a set

1. Suppose $R \in R$, but by definition $R \not\in R$, contradiction.

2. Suppose $R \not\in R$... (I am not sure what should be the contradiction here)

My guess: then $\{S | S \not\in S\} \not\in R$, then $R$ is not the set of all sets such that $S \not\in S$ OR $R$ is empty, but clearly it cannot be empty (?)

Can someone point out the contradiction in the second one?

Answer 1

Suppose $R\not \in R$. Then $R$ is a set that does not have itself as an element.

Since $R$ satisfies this condition, we conclude $R\in R$.... woops.