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Definition df-pw 3640
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 20832). We will later introduce the Axiom of Power Sets ax-pow 4204, which can be expressed in class notation per pwexg 4210. Still later we will prove, in hashpw 11404, 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 3638 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1631 . . . 4  class  x
54, 1wss 3165 . . 3  wff  x  C_  A
65, 3cab 2282 . 2  class  { x  |  x  C_  A }
72, 6wceq 1632 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3641  elpw  3644  nfpw  3649  pwss  3652  pw0  3778  pwpw0  3779  snsspw  3800  pwsn  3837  pwsnALT  3838  pwex  4209  abssexg  4211  iunpw  4586  orduniss2  4640  mapex  6794  ssenen  7051  domtriomlem  8084  npex  8626  isbasis2g  16702  avril1  20852  dfon2lem2  24211  psubspset  30555  psubclsetN  30747
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