Definition df-pw 4624

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 30461). We will later introduce the Axiom of Power Sets ax-pow 5383, which can be expressed in class notation per pwexg 5396. Still later we will prove, in hashpw 14485, 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 4622	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1536	. . . 4 class 𝑥
5	4, 1	wss 3976	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2717	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1537	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4625  pweqALT  4637  nfpw  4641  pw0  4837  pwpw0  4838  pwsn  4924  vpwex  5395  abssexg  5400  orduniss2  7869  mapexOLD  8890  ssenen  9217  domtriomlem  10511  npex  11055  ustval  24232  avril1  30495  fineqvpow  35072  dfon2lem2  35748  bj-velpwALT  37019