Definition df-pw 4541

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 28208). We will later introduce the Axiom of Power Sets ax-pow 5266, which can be expressed in class notation per pwexg 5279. Still later we will prove, in hashpw 13798, 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 4539	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1536	. . . 4 class 𝑥
5	4, 1	wss 3936	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2799	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1537	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4542  elpwOLD  4545  pweqALT  4556  nfpw  4560  pw0  4745  pwpw0  4746  pwsn  4830  pwsnOLD  4831  vpwex  5278  abssexg  5283  orduniss2  7548  mapex  8412  ssenen  8691  domtriomlem  9864  npex  10408  ustval  22811  avril1  28242  dfon2lem2  33029