# Metamath Proof Explorer

## Definition df-pw

Description: Define power class. Definition 5.10 of TakeutiZaring p. 17, but we also let it apply to proper classes, i.e. those that are not members of _V . When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if A = { 3 , 5 , 7 } , then ~P A = { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } } ( ex-pw ). We will later introduce the Axiom of Power Sets ax-pow , which can be expressed in class notation per pwexg . Still later we will prove, in hashpw , that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 24-Jun-1993)

Ref Expression
Assertion df-pw ${⊢}𝒫{A}=\left\{{x}|{x}\subseteq {A}\right\}$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ${class}{A}$
1 0 cpw ${class}𝒫{A}$
2 vx ${setvar}{x}$
3 2 cv ${setvar}{x}$
4 3 0 wss ${wff}{x}\subseteq {A}$
5 4 2 cab ${class}\left\{{x}|{x}\subseteq {A}\right\}$
6 1 5 wceq ${wff}𝒫{A}=\left\{{x}|{x}\subseteq {A}\right\}$