Does this Set Represent a Natural Number? In set theory, the natural numbers \$\mathbb{N} = \{0, 1, 2, 3, ...\}\$ are usually encoded as pure sets, that is sets which only contain the empty set or other sets that are pure. However, not all pure sets represent natural numbers. This challenge is about deciding whether a given pure set represents an encoding of natural number or not.
The encoding of natural numbers works in the following way1:

Zero is the empty set: \$ \text{Set}(0) = \{\} \$
For a number \$n &gt; 0\$: \$ \text{Set}(n) = \text{Set}(n-1) \cup \{\text{Set}(n-1)\}\$

Thus, the encodings of the first few natural numbers are

\$ 0 \leadsto \{\}\$
\$ 1 \leadsto \{0\} \leadsto \{\{\}\}\$
\$ 2 \leadsto \{0,1\} \leadsto \{\{\},\{\{\}\}\}\$
\$ 3 \leadsto \{0,1,2\} \leadsto \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\$
\$ 4 \leadsto \{0,1,2,3\} \leadsto \{\{\},\{\{\}\},\{\{\},\{\{\}\}\},\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\}\$

The Task

Given a string representing a pure set, determine whether this set encodes a natural number according to the above construction.
Note, however, that the elements of a set are not ordered, so \$\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\$ is not the only valid representation of \$3\$ as e.g. \$\{\{\{\}\},\{\},\{\{\{\}\},\{\}\}\}\$ represents the same set.
You may use [], () or &lt;&gt; instead of {}.
You may assume the sets are given without the , as separator.
You can assume there won't be any duplicate elements in the input, e.g. {{},{}} is not a valid input, and that the input is well-formed, e.g. no {{},, {,{}} or similar.

Test Cases
True:
{}
{{}}
{{},{{}}}
{{{}},{}}
{{},{{}},{{},{{}}}}
{{{},{{}}},{},{{}}}
{{{{}},{}},{{}},{}}
{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}
{{{{{}},{}},{{}},{}},{{}},{},{{},{{}}}}
{{},{{}},{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}},{{{}},{}},{{},{{}},{{},{{}}}}}
{{{{{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}}

False:
{{{}}}
{{{{}}}}
{{{{}},{}}}
{{},{{}},{{{}}}}
{{{},{{}}},{{}}}
{{{{{}}},{}},{{}},{}}
{{},{{}},{{},{{}}},{{},{{}},{{{}}}}}
{{{{{}},{}},{{{}}},{}},{{}},{},{{},{{}}}}
{{{{{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}},{{{{{}},{}},{{}},{}},{{{}},{}},{{}}},{{{{}},{}},{{}},{}},{{{}},{}},{{}},{}}


Related: Natural Construction (Output the set encoding of a given natural number.)
1 See https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
JavaScript (Node.js), 53 48 44 bytes



f=a=&gt;(a=eval(a)).every(e=&gt;a[e.length]&amp;&amp;f(e))


Try it online! Test cases mostly shamelessly stolen from @Arnauld's answer. Explanation: If a set represents a natural number, then the natural number it represents must be equal to the size of the set, and (given that the elements are guaranteed distinct) the elements must be the representations of the natural numbers less than it, and these must therefore have shorter lengths. This is trivially true of the empty set of course. Edit: Saved 5 bytes thanks to @Arnauld. Saved 4 bytes thanks to @Cowsquack.