Definition df-pw 4534

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 28800). We will later introduce the Axiom of Power Sets ax-pow 5287, which can be expressed in class notation per pwexg 5300. Still later we will prove, in hashpw 14158, 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 4532	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1537	. . . 4 class 𝑥
5	4, 1	wss 3886	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2712	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1538	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4535  elpwOLD  4538  pweqALT  4549  nfpw  4553  pw0  4744  pwpw0  4745  pwsn  4830  pwsnOLD  4831  vpwex  5299  abssexg  5304  orduniss2  7687  mapex  8628  ssenen  8945  domtriomlem  10205  npex  10749  ustval  23361  avril1  28834  fineqvpow  33072  dfon2lem2  33767