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Definition df-pw 4499
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 28214). We will later introduce the Axiom of Power Sets ax-pow 5231, which can be expressed in class notation per pwexg 5244. Still later we will prove, in hashpw 13793, 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 4497 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1537 . . . 4 class 𝑥
54, 1wss 3881 . . 3 wff 𝑥𝐴
65, 3cab 2776 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1538 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  elpwg  4500  elpwOLD  4503  pweqALT  4514  nfpw  4518  pw0  4705  pwpw0  4706  pwsn  4792  pwsnOLD  4793  vpwex  5243  abssexg  5248  orduniss2  7528  mapex  8395  ssenen  8675  domtriomlem  9853  npex  10397  ustval  22808  avril1  28248  dfon2lem2  33142
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