Definition df-pw 4602

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 30448). We will later introduce the Axiom of Power Sets ax-pow 5365, which can be expressed in class notation per pwexg 5378. Still later we will prove, in hashpw 14475, 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 4600	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1539	. . . 4 class 𝑥
5	4, 1	wss 3951	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2714	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1540	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4603  pweqALT  4615  nfpw  4619  pw0  4812  pwpw0  4813  pwsn  4900  vpwex  5377  abssexg  5382  orduniss2  7853  mapexOLD  8872  ssenen  9191  domtriomlem  10482  npex  11026  ustval  24211  avril1  30482  fineqvpow  35110  dfon2lem2  35785  bj-velpwALT  37054