MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-pw Unicode version

Definition df-pw 3532
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 20629). We will later introduce the Axiom of Power Sets ax-pow 4082, which can be expressed in class notation per pwexg 4088. Still later we will prove, in hashpw 11265, 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  =  { x  |  x 
C_  A }
Distinct variable group:    x, A

Detailed syntax breakdown of Definition df-pw
StepHypRef Expression
1 cA . . 3  class  A
21cpw 3530 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1618 . . . 4  class  x
54, 1wss 3078 . . 3  wff  x  C_  A
65, 3cab 2239 . 2  class  { x  |  x  C_  A }
72, 6wceq 1619 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3533  elpw  3536  nfpw  3540  pwss  3543  pw0  3662  pwpw0  3663  snsspw  3684  pwsn  3721  pwsnALT  3722  pwex  4087  abssexg  4089  iunpw  4461  orduniss2  4515  mapex  6664  ssenen  6920  domtriomlem  7952  npex  8490  isbasis2g  16518  avril1  20666  dfon2lem2  23308  psubspset  28622  psubclsetN  28814
  Copyright terms: Public domain W3C validator