Proving $\{a\}$ is a set

Question

I recently came across the following post Proving $\{a\}$ is a also set given that $a$ is a set. Introduction to Set Theory.. While I understand the answer given I am still unsure whether or not the proof given in the OP is correct, i.e. if it is legitimate to use the axiom of extensionality. It seems to me that the answer given only suggests that there is no need to use the axiom of extensionality, not that this is not permitted. Any help would be appreciated. Many thanks!

Answer 1

A set can be defined in three ways:

1- By extension

2- By comprehension

3- By Venn diagram.

In your case $ \{a\} $ is a set defined by extension. it contains only one element which is $ a$.

In set Theory, same objects can be seen as sets or as elements.

$$\{a\} \text{ is an element of the set } \{\{a\}\}$$