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Definition df-pw 4379
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 27843). We will later introduce the Axiom of Power Sets ax-pow 5064, which can be expressed in class notation per pwexg 5077. Still later we will prove, in hashpw 13511, 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 4377 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1657 . . . 4 class 𝑥
54, 1wss 3797 . . 3 wff 𝑥𝐴
65, 3cab 2810 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1658 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  4380  elpw  4383  nfpw  4391  pw0  4560  pwpw0  4561  pwsn  4649  pwsnALT  4650  vpwex  5076  abssexg  5080  orduniss2  7293  mapex  8127  ssenen  8402  domtriomlem  9578  npex  10122  ustval  22375  avril1  27876  dfon2lem2  32226
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