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Definition df-pw 3627
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 20816). We will later introduce the Axiom of Power Sets ax-pow 4188, which can be expressed in class notation per pwexg 4194. Still later we will prove, in hashpw 11388, 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 3625 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1622 . . . 4  class  x
54, 1wss 3152 . . 3  wff  x  C_  A
65, 3cab 2269 . 2  class  { x  |  x  C_  A }
72, 6wceq 1623 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3628  elpw  3631  nfpw  3636  pwss  3639  pw0  3762  pwpw0  3763  snsspw  3784  pwsn  3821  pwsnALT  3822  pwex  4193  abssexg  4195  iunpw  4570  orduniss2  4624  mapex  6778  ssenen  7035  domtriomlem  8068  npex  8610  isbasis2g  16686  avril1  20836  dfon2lem2  24140  psubspset  29933  psubclsetN  30125
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