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Definition df-pw 4292
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 27568). We will later introduce the Axiom of Power Sets ax-pow 4980, which can be expressed in class notation per pwexg 4987. Still later we will prove, in hashpw 13386, 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 4290 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1619 . . . 4 class 𝑥
54, 1wss 3703 . . 3 wff 𝑥𝐴
65, 3cab 2734 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1620 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  4293  elpw  4296  nfpw  4304  pw0  4476  pwpw0  4477  pwsn  4568  pwsnALT  4569  pwex  4985  abssexg  4988  orduniss2  7186  mapex  8017  ssenen  8287  domtriomlem  9427  npex  9971  ustval  22178  avril1  27601  dfon2lem2  31965
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