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  1. By axiom of infinity, we have a set.
  2. By axiom of specification, we can create all of its subsets.
  3. By axiom of pairing and axiom of unions, we can create a set containing all of theses created subsets.

Example: Suppose a set $S$ has $a,b$ and $c$ as its only distinct subsets. (Well it isn’t possible, but for the time’s sake let it be. It is also applicable for correct number of subsets.) Now, by pairing we have $\{ a,b\}$ and $\{ c\}$. By pairing again, we have $\{ \{ a,b\} ,\{ c\} \}$. By union, we have $\{ a,b,c\}$.

We have thus constructed the power set of the given set.

Why then do we need axiom of powers?

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  • There are no sets with exactly three subsets. Also, how do you deal with infinitely many subsets? – Asaf Karagila Jun 28 '19 at 08:39
  • Maybe also of interest: https://math.stackexchange.com/questions/1446509/zf-difference-between-pow-and-sep, https://math.stackexchange.com/questions/1619210/why-do-we-want-the-axiom-of-the-power-set, and I think I missed a few. – Asaf Karagila Jun 28 '19 at 08:46

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