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# Transitive subsets

#### Description

Let $$V_0 = \varnothing$$ be the empty set and for $$i>0$$, let $$V_i = \mathcal{P}(V_{i-1})$$, where $$\mathcal{P}(X)$$ is the set of all subsets of $$X$$ (also known as the power set).  So:

$$V_1=\{\varnothing\}$$

$$V_2 = \{\varnothing,\{\varnothing\}\}$$

$$V_3=\{\varnothing,\{\varnothing\},\{\{\varnothing\}\},\{\varnothing,\{\varnothing\}\}\}$$

and so on.

A set $$X$$ is called transitive if each element of $$X$$ is also a subset of $$X$$.

#### Question

How many elements in $$V_5$$ are transitive?

How many elements in $$V_5$$ are transitive?