I've been thinking about the Cantor set lately and I'm struggling with the proof of the obvious fact that the Cantor set is self-similar. This question Show that the Cantor set is self-similar and the accepted answer were very useful. However, I lack the other direction of the proof. How to show that if $c\in C$, then $c\in \frac{1}{3}C$ or $c\in(\frac{2}{3}+\frac{1}{3}C)$?
Definitions and the other direction of the proof can be read from the link. Btw. there isn't a tag for self-similarity.
Edit: this book seems to have a complete proof. It requires the result which says that the Cantor set contains only numbers which can be expressed in ternary with only zeros and twos, tough. I would still like to see a set-theoretic proof if there is any.
Edit 2: this book has the approach I was looking for.