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Definition df-pw 3737
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 21578). We will later introduce the Axiom of Power Sets ax-pow 4311, which can be expressed in class notation per pwexg 4317. Still later we will prove, in hashpw 11619, 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 3735 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1648 . . . 4  class  x
54, 1wss 3256 . . 3  wff  x  C_  A
65, 3cab 2366 . 2  class  { x  |  x  C_  A }
72, 6wceq 1649 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3738  elpw  3741  nfpw  3746  pwss  3749  pw0  3881  pwpw0  3882  snsspw  3905  pwsn  3944  pwsnALT  3945  pwex  4316  abssexg  4318  iunpw  4692  orduniss2  4746  mapex  6953  ssenen  7210  domtriomlem  8248  npex  8789  isbasis2g  16929  ustval  18146  avril1  21598  dfon2lem2  25157  psubspset  29909  psubclsetN  30101
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