Why isn't $\{\{\emptyset\}\} = \{\emptyset,\{\emptyset\}\}$? If the set $ \{\emptyset,\{\emptyset\}\}$ has just one element that is $\{\emptyset\}$ and is empty otherwise, shouldn't it be equivalent to $\{\{\emptyset\}\}$?
 A: Assuming your "$\Phi$" is the empty set $\{\}$ (usually denoted "$\emptyset$," LaTeX code "\$\emptyset\$"), the important point is that $\emptyset$ is not nothing. It contains nothing, but that's not the same thing, any more than an empty bag is the same as no bag at all.
It's important to say at this point that you shouldn't push the bag metaphor too far, but in some contexts - including this one - I think it is useful.
In particular, $\{\emptyset,\{\emptyset\}\}$ has two elements - we can't ignore the by-itself $\emptyset$. After all, if we could the whole thing would evaporate: highlighting in red the bits that we erase at each step we'd get $$\mbox{$\{\color{red}{\emptyset},\{\color{red}{\emptyset}\}\}=\{\color{red}{\{\}}\}=\color{red}{\{\}}=\quad$ .}$$ I'm not even sure what that last thing is!
It's worth mentioning at this point that the usual framework of set theory builds everything up from the emptyset alone. So far from being a silly hair-splitting, caution around the emptyset is quite serious mathematics.
