Definition df-pw 4565

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 30358). We will later introduce the Axiom of Power Sets ax-pow 5320, which can be expressed in class notation per pwexg 5333. Still later we will prove, in hashpw 14401, 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 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

Distinct variable group:   𝑥,𝐴

Detailed syntax breakdown of Definition df-pw

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	cpw 4563	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1539	. . . 4 class 𝑥
5	4, 1	wss 3914	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2707	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1540	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4566  pweqALT  4578  nfpw  4582  pw0  4776  pwpw0  4777  pwsn  4864  vpwex  5332  abssexg  5337  orduniss2  7808  mapexOLD  8805  ssenen  9115  domtriomlem  10395  npex  10939  ustval  24090  avril1  30392  fineqvpow  35086  dfon2lem2  35772  bj-velpwALT  37041