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 = { x | x C_ A }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA
 |-  A
1 0 cpw
 |-  ~P A
2 vx
 |-  x
3 2 cv
 |-  x
4 3 0 wss
 |-  x C_ A
5 4 2 cab
 |-  { x | x C_ A }
6 1 5 wceq
 |-  ~P A = { x | x C_ A }