Theorem. The set of all subsets of a set is a set.
Proof. Given an element of the power set $f \in 2^X$, a subset $y \subseteq X$ and a statement $P(f,y)$ $$P(f,y) = (f^{-1}(\{1\}) = y)$$ such that for every object $f$ there is at most one object $y$ resulting in $P(f,y)$ to be true. Then, according to the replacement axiom (3.6), there exists a set $\{y : P(f,y) \: \text{for some} \: f \in 2^X \}$ such that for any object $z$ $$z \in \{y : P(f,y) \: \text{for some} \: f \in 2^X \} \leftrightarrow P(f,z) \: \text{for some} \: f \in 2^X,$$ where $\{y : P(f,y) \: \text{for some} \: f \in 2^X \} = \{ y : y \: \text{is a subset of} \: X \}. \Box$

2015 DEC 04 (v1.0)
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