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Definition df-pw 3825
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 21768). We will later introduce the Axiom of Power Sets ax-pow 4406, which can be expressed in class notation per pwexg 4412. Still later we will prove, in hashpw 11730, 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 3823 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1652 . . . 4  class  x
54, 1wss 3306 . . 3  wff  x  C_  A
65, 3cab 2428 . 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  3826  elpw  3829  nfpw  3834  pwss  3837  pw0  3969  pwpw0  3970  snsspw  3994  pwsn  4033  pwsnALT  4034  pwex  4411  abssexg  4413  iunpw  4788  orduniss2  4842  mapex  7053  ssenen  7310  domtriomlem  8353  npex  8894  isbasis2g  17044  ustval  18263  avril1  21788  dfon2lem2  25442  psubspset  30639  psubclsetN  30831
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