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Definition df-pw 4013
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 26417). We will later introduce the Axiom of Power Sets ax-pow 4668, which can be expressed in class notation per pwexg 4675. Still later we will prove, in hashpw 12946, 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 4011 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1473 . . . 4 class 𝑥
54, 1wss 3444 . . 3 wff 𝑥𝐴
65, 3cab 2500 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1474 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
This definition is referenced by:  pweq  4014  elpw  4017  nfpw  4023  pw0  4186  pwpw0  4187  pwsn  4264  pwsnALT  4265  pwex  4673  abssexg  4676  orduniss2  6800  mapex  7625  ssenen  7894  domtriomlem  9022  npex  9562  ustval  21717  avril1  26450  dfon2lem2  30776
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