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Definition df-pw 3761
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 21690). We will later introduce the Axiom of Power Sets ax-pow 4337, which can be expressed in class notation per pwexg 4343. Still later we will prove, in hashpw 11654, 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 3759 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1648 . . . 4  class  x
54, 1wss 3280 . . 3  wff  x  C_  A
65, 3cab 2390 . 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  3762  elpw  3765  nfpw  3770  pwss  3773  pw0  3905  pwpw0  3906  snsspw  3930  pwsn  3969  pwsnALT  3970  pwex  4342  abssexg  4344  iunpw  4718  orduniss2  4772  mapex  6983  ssenen  7240  domtriomlem  8278  npex  8819  isbasis2g  16968  ustval  18185  avril1  21710  dfon2lem2  25354  psubspset  30226  psubclsetN  30418
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