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Definition df-pw 3803
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 21742). We will later introduce the Axiom of Power Sets ax-pow 4380, which can be expressed in class notation per pwexg 4386. Still later we will prove, in hashpw 11704, 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 3801 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1652 . . . 4  class  x
54, 1wss 3322 . . 3  wff  x  C_  A
65, 3cab 2424 . 2  class  { x  |  x  C_  A }
72, 6wceq 1653 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3804  elpw  3807  nfpw  3812  pwss  3815  pw0  3947  pwpw0  3948  snsspw  3972  pwsn  4011  pwsnALT  4012  pwex  4385  abssexg  4387  iunpw  4762  orduniss2  4816  mapex  7027  ssenen  7284  domtriomlem  8327  npex  8868  isbasis2g  17018  ustval  18237  avril1  21762  dfon2lem2  25416  psubspset  30615  psubclsetN  30807
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