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Definition df-pw 4109
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 26444). We will later introduce the Axiom of Power Sets ax-pow 4764, which can be expressed in class notation per pwexg 4771. Still later we will prove, in hashpw 13035, 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 4107 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1473 . . . 4 class 𝑥
54, 1wss 3539 . . 3 wff 𝑥𝐴
65, 3cab 2595 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1474 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  4110  elpw  4113  nfpw  4119  pw0  4282  pwpw0  4283  pwsn  4360  pwsnALT  4361  pwex  4769  abssexg  4772  orduniss2  6902  mapex  7727  ssenen  7996  domtriomlem  9124  npex  9664  ustval  21758  avril1  26477  dfon2lem2  30739
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