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Definition df-pw 3417
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 3415 . 2  class  ~P A
3 vx . . . . 5  setvar  x
43cv 1286 . . . 4  class  x
54, 1wss 2988 . . 3  wff  x  C_  A
65, 3cab 2071 . 2  class  { x  |  x  C_  A }
72, 6wceq 1287 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3418  elpw  3421  nfpw  3427  pwss  3430  pw0  3569  snsspw  3593  pwsnss  3632  vpwex  3991  abssexg  3993  iunpw  4277  iotass  4965  mapex  6365  ssenen  6521  bdcpw  11229
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