Cardinality of Power Set of Finite Set/Informal Proof

Proof
Given an element $x$ of $S$, each subset of $S$ either includes $x$ or does not include $x$ (this follows directly from the definition of a set), which gives us two possibilities.

The same reasoning holds for any element of $S$.

One can intuitively see that this means that there are $\displaystyle \underbrace{2 \times 2 \times \ldots \times 2}_{\left|{S}\right|} = 2^{\left|{S}\right|}$ total possible combinations of elements of $S$.

This is exactly $\left|{\mathcal P \left({S}\right)}\right|$.

Note
The formal mathematical backing for the intuitive leap made in this "proof" is non-trivial, so while this it serves as an excellent demonstration of why this result holds true, it does not constitute a fully rigorous proof of this theorem.