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Definition df-pw 3587
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 20745). We will later introduce the Axiom of Power Sets ax-pow 4146, which can be expressed in class notation per pwexg 4152. Still later we will prove, in hashpw 11339, 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 3585 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1618 . . . 4  class  x
54, 1wss 3113 . . 3  wff  x  C_  A
65, 3cab 2242 . 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  3588  elpw  3591  nfpw  3596  pwss  3599  pw0  3722  pwpw0  3723  snsspw  3744  pwsn  3781  pwsnALT  3782  pwex  4151  abssexg  4153  iunpw  4528  orduniss2  4582  mapex  6732  ssenen  6989  domtriomlem  8022  npex  8564  isbasis2g  16634  avril1  20782  dfon2lem2  23495  psubspset  29084  psubclsetN  29276
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