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Definition df-pw 3408
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 is { 3 , 5 , 7 }, then 
~P A is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove 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 3406 . 2  class  ~P A
3 vx . . . . 5  setvar  x
43cv 1284 . . . 4  class  x
54, 1wss 2984 . . 3  wff  x  C_  A
65, 3cab 2069 . 2  class  { x  |  x  C_  A }
72, 6wceq 1285 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3409  elpw  3412  nfpw  3418  pwss  3421  pw0  3558  snsspw  3582  pwsnss  3621  pwex  3979  abssexg  3981  iunpw  4264  iotass  4950  mapex  6340  ssenen  6496  bdcpw  11102
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