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Definition df-pw 4151
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 27256). We will later introduce the Axiom of Power Sets ax-pow 4834, which can be expressed in class notation per pwexg 4841. Still later we will prove, in hashpw 13206, 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 4149 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1480 . . . 4 class 𝑥
54, 1wss 3567 . . 3 wff 𝑥𝐴
65, 3cab 2606 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1481 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  4152  elpw  4155  nfpw  4163  pw0  4334  pwpw0  4335  pwsn  4419  pwsnALT  4420  pwex  4839  abssexg  4842  orduniss2  7018  mapex  7848  ssenen  8119  domtriomlem  9249  npex  9793  ustval  21987  avril1  27289  dfon2lem2  31663
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