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Definition df-pw 3980
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 26039). We will later introduce the Axiom of Power Sets ax-pow 4619, which can be expressed in class notation per pwexg 4626. Still later we will prove, in hashpw 12728, 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 3978 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1467 . . . 4 class 𝑥
54, 1wss 3426 . . 3 wff 𝑥𝐴
65, 3cab 2491 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1468 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  3981  elpw  3984  nfpw  3990  pw0  4148  pwpw0  4149  pwsn  4222  pwsnALT  4223  pwex  4624  abssexg  4627  orduniss2  6737  mapex  7561  ssenen  7830  domtriomlem  8957  npex  9496  ustval  21375  avril1  26061  dfon2lem2  30581
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