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Definition df-pw 4110
 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 26678). We will later introduce the Axiom of Power Sets ax-pow 4769, which can be expressed in class notation per pwexg 4776. Still later we will prove, in hashpw 13083, 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 4108 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1474 . . . 4 class 𝑥
54, 1wss 3540 . . 3 wff 𝑥𝐴
65, 3cab 2596 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1475 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
 Colors of variables: wff setvar class This definition is referenced by:  pweq  4111  elpw  4114  nfpw  4120  pw0  4283  pwpw0  4284  pwsn  4366  pwsnALT  4367  pwex  4774  abssexg  4777  orduniss2  6925  mapex  7750  ssenen  8019  domtriomlem  9147  npex  9687  ustval  21816  avril1  26711  dfon2lem2  30933
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