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Definition df-pw 4299
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 27621). We will later introduce the Axiom of Power Sets ax-pow 4974, which can be expressed in class notation per pwexg 4980. Still later we will prove, in hashpw 13418, 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 4297 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1630 . . . 4 class 𝑥
54, 1wss 3723 . . 3 wff 𝑥𝐴
65, 3cab 2757 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1631 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  4300  elpw  4303  nfpw  4311  pw0  4478  pwpw0  4479  pwsn  4566  pwsnALT  4567  vpwex  4979  abssexg  4982  orduniss2  7178  mapex  8013  ssenen  8288  domtriomlem  9464  npex  10008  ustval  22219  avril1  27654  dfon2lem2  32018
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