This is exercise $3.1.2$ on Tao's Analysis I and the exact wording is.
Exercise $3.1.2.$ Using only Axiom 3.2, Axiom 3.1, Axiom 3.3, and Axiom 3.4, prove that the sets
∅, {∅}, {{∅}}, and {∅, {∅}} are all distinct (i.e., no two of them are equal to each other).
The axioms are
Axiom 3.1 (Sets are objects). If $A$ is a set, then $A$ is also an object. In particular,
given two sets $A$ and $B$, it is meaningful to ask whether $A$ is also an element of $B$.
Axiom 3.2 (Equality of sets). Two sets $A$ and $B$ are equal, $A = B$, iff every element
of $A$ is an element of $B$ and vice versa. To put it another way, $A = B$ if and only if
every element $x$ of $A$ belongs also to $B$, and every element $y$ of $B$ belongs also to $A$.
Axiom 3.3 (Empty set). There exists a set $∅$, known as the empty set, which contains
no elements, i.e., for every object $x$ we have $x \notin ∅$.
Axiom 3.4 (Singleton sets and pair sets). If $a$ is an object, then there exists a set
$\{a\}$ whose only element is $a$, i.e., for every object $y$, we have $y \in \{a\}$ if and only if
$y = a$; we refer to $\{a\}$ as the singleton set whose element is $a$. Furthermore, if $a$ and
$b$ are objects, then there exists a set $\{a, b\}$ whose only elements are $a$ and $b$; i.e., for every object $y$, we have $y \in \{a, b\}$ if and only if $y = a $ or $y = b$; we refer to this set as the pair set formed by $a$ and $b$.
I can intuitively see how these are not equal to each other since the empty set has no elements and the others have elements and then none of the other sets obey Axiom 3.2. But still, I struggled formulating a rigorous proof.
All of these sets are also objects. Suppose that $\{\emptyset\}$, $\{\{\emptyset\}\}$, and $\{\emptyset,\{\emptyset\}\}$ are equal to the empty set. For every object $x$ we should have $x \notin \emptyset$ but since $\emptyset \in \{\emptyset\}$, $\{\emptyset\}\in \{\{\emptyset\}\}$, and $\emptyset \in \{\emptyset,\{\emptyset\}\}$ we get contradictions all around, proving $\emptyset$ is different from the other three. For the singleton set $\{\emptyset\}$ for every object $y$ we have $y \in \{\emptyset\} \iff y=\emptyset$ but since we've shown that $\emptyset$ is different from the other three sets, $y$ is different from the other sets by the axiom of substitution so $\{\emptyset\}$ is different from the other three sets. We can repeat this process for $\{\{\emptyset\}\}$. Which would leave the last set to be different from the other sets as well.
The fact that I was allowed to say that last sentence makes me believe that this proof is wrong. I can't put my finger on why but the whole proof looks very sketchy.
Edit: To clarify, this was an attempt at a faster proof which didn’t work out. I’m not asking for another standard proof but I didn’t make that clear from the beginning. In my mind it made sense that a faster proof was possible but I couldn’t come up with it in the end. For a proof of this, you can look at the answers.
Best Answer
To show that $\varnothing$ and $\{\varnothing\}$ are distinct, by Axiom 3.3, $\varnothing$ denotes the empty set which contains no elements. By Axiom 3.1, since $\varnothing$ is a set, it is also an object. Further, $\{\varnothing\}$ contains the object $\varnothing$. Thus, since one set contains no elements and another contains an element, by Axiom 3.2, the sets are not equal.
By a similar argument $\varnothing$ is distinct from the other three sets, as those contain elements.
To show that $\{\varnothing\}$ and $\{\{\varnothing\}\}$ are distinct, we refer to Axiom 3.2 again. We compare the elements in each set and we find that $\varnothing$ is not equal to $\{\varnothing\}$ by the argument we already made. So the sets $\{\varnothing\}$ and $\{\{\varnothing\}\}$ are different because each set contains an element that the other does not.
Finally, $\{\varnothing, \{\varnothing\}\}$ is distinct from both $\{\varnothing\}$ and $\{\{\varnothing\}\}$ because it contains two elements while the other sets contain only one element (Axiom 3.2)
I suppose also that Axiom 3.4 is used to justify the existence of the singleton and pair sets.