MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-pw Unicode version

Definition df-pw 3769
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 21698). We will later introduce the Axiom of Power Sets ax-pow 4345, which can be expressed in class notation per pwexg 4351. Still later we will prove, in hashpw 11662, 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 3767 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1648 . . . 4  class  x
54, 1wss 3288 . . 3  wff  x  C_  A
65, 3cab 2398 . 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  3770  elpw  3773  nfpw  3778  pwss  3781  pw0  3913  pwpw0  3914  snsspw  3938  pwsn  3977  pwsnALT  3978  pwex  4350  abssexg  4352  iunpw  4726  orduniss2  4780  mapex  6991  ssenen  7248  domtriomlem  8286  npex  8827  isbasis2g  16976  ustval  18193  avril1  21718  dfon2lem2  25362  psubspset  30238  psubclsetN  30430
  Copyright terms: Public domain W3C validator