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

Definition df-pw 4539
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 28136). We will later introduce the Axiom of Power Sets ax-pow 5258, which can be expressed in class notation per pwexg 5271. Still later we will prove, in hashpw 13787, 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, 24-Jun-1993.)
Assertion
Ref Expression
df-pw 𝒫 𝐴 = {𝑥𝑥𝐴}
Distinct variable group:   𝑥,𝐴

Detailed syntax breakdown of Definition df-pw
StepHypRef Expression
1 cA . . 3 class 𝐴
21cpw 4537 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1527 . . . 4 class 𝑥
54, 1wss 3935 . . 3 wff 𝑥𝐴
65, 3cab 2799 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1528 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  4540  elpwg  4543  elpwOLD  4546  nfpw  4553  pw0  4739  pwpw0  4740  pwsn  4824  pwsnALT  4825  vpwex  5270  abssexg  5274  orduniss2  7536  mapex  8402  ssenen  8680  domtriomlem  9853  npex  10397  ustval  22740  avril1  28170  dfon2lem2  32927
  Copyright terms: Public domain W3C validator