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Definition df-pw 3793
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 21729). We will later introduce the Axiom of Power Sets ax-pow 4369, which can be expressed in class notation per pwexg 4375. Still later we will prove, in hashpw 11691, 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 3791 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1651 . . . 4  class  x
54, 1wss 3312 . . 3  wff  x  C_  A
65, 3cab 2421 . 2  class  { x  |  x  C_  A }
72, 6wceq 1652 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3794  elpw  3797  nfpw  3802  pwss  3805  pw0  3937  pwpw0  3938  snsspw  3962  pwsn  4001  pwsnALT  4002  pwex  4374  abssexg  4376  iunpw  4751  orduniss2  4805  mapex  7016  ssenen  7273  domtriomlem  8314  npex  8855  isbasis2g  17005  ustval  18224  avril1  21749  dfon2lem2  25403  psubspset  30478  psubclsetN  30670
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