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

Definition df-pw 3629
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 20792). 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 11383, 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 3627 . 2  class  ~P A
3 vx . . . . 5  set  x
43cv 1623 . . . 4  class  x
54, 1wss 3154 . . 3  wff  x  C_  A
65, 3cab 2271 . 2  class  { x  |  x  C_  A }
72, 6wceq 1624 1  wff  ~P A  =  { x  |  x 
C_  A }
Colors of variables: wff set class
This definition is referenced by:  pweq  3630  elpw  3633  nfpw  3638  pwss  3641  pw0  3764  pwpw0  3765  snsspw  3786  pwsn  3823  pwsnALT  3824  pwex  4193  abssexg  4195  iunpw  4570  orduniss2  4624  mapex  6774  ssenen  7031  domtriomlem  8064  npex  8606  isbasis2g  16681  avril1  20829  dfon2lem2  23542  psubspset  29201  psubclsetN  29393
  Copyright terms: Public domain W3C validator