Definition df-pw 4562

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 31057). We will later introduce the Axiom of Power Sets ax-pow 5331, which can be expressed in class notation per pwexg 5344. Still later we will prove, in hashpw 14605, 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 4560	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1569	. . . 4 class 𝑥
5	4, 1	wss 3900	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2741	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1570	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is used by:  elpwg  4563  pweqALT  4575  nfpw  4580  pwsn  4830  pw0  4833  pwpw0  4834  vpwex  5343  abssexg  5348  orduniss2  7865  ssenen  9195  domtriomlem  10544  npex  11095  ustval  24549  avril1  31091  fineqvpow  35882  dfon2lem2  36723  bj-velpwALT  38158