Definition df-pw 4501

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 28466). We will later introduce the Axiom of Power Sets ax-pow 5243, which can be expressed in class notation per pwexg 5256. Still later we will prove, in hashpw 13968, 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 4499	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1542	. . . 4 class 𝑥
5	4, 1	wss 3853	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2714	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1543	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4502  elpwOLD  4505  pweqALT  4516  nfpw  4520  pw0  4711  pwpw0  4712  pwsn  4797  pwsnOLD  4798  vpwex  5255  abssexg  5260  orduniss2  7590  mapex  8492  ssenen  8798  domtriomlem  10021  npex  10565  ustval  23054  avril1  28500  fineqvpow  32732  dfon2lem2  33430