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Definition df-pw 4132
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 27140). We will later introduce the Axiom of Power Sets ax-pow 4803, which can be expressed in class notation per pwexg 4810. Still later we will prove, in hashpw 13163, 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 4130 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1479 . . . 4 class 𝑥
54, 1wss 3555 . . 3 wff 𝑥𝐴
65, 3cab 2607 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1480 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  4133  elpw  4136  nfpw  4143  pw0  4311  pwpw0  4312  pwsn  4396  pwsnALT  4397  pwex  4808  abssexg  4811  orduniss2  6980  mapex  7808  ssenen  8078  domtriomlem  9208  npex  9752  ustval  21916  avril1  27173  dfon2lem2  31387
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