Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-pw Structured version   GIF version

Definition df-pw 3353
 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 is { 3 , 5 , 7 }, then 𝒫 A is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove 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, 5-Aug-1993.)
Assertion
Ref Expression
df-pw 𝒫 A = {xxA}
Distinct variable group:   x,A

Detailed syntax breakdown of Definition df-pw
StepHypRef Expression
1 cA . . 3 class A
21cpw 3351 . 2 class 𝒫 A
3 vx . . . . 5 setvar x
43cv 1241 . . . 4 class x
54, 1wss 2911 . . 3 wff xA
65, 3cab 2023 . 2 class {xxA}
72, 6wceq 1242 1 wff 𝒫 A = {xxA}
 Colors of variables: wff set class This definition is referenced by:  pweq  3354  elpw  3357  nfpw  3363  pwss  3366  pw0  3502  snsspw  3526  pwsnss  3565  pwex  3923  abssexg  3925  iunpw  4177  iotass  4827  bdcpw  9324
 Copyright terms: Public domain W3C validator