Definition df-pw 4532

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 28694). We will later introduce the Axiom of Power Sets ax-pow 5283, which can be expressed in class notation per pwexg 5296. Still later we will prove, in hashpw 14079, 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 4530	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1538	. . . 4 class 𝑥
5	4, 1	wss 3883	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2715	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1539	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4533  elpwOLD  4536  pweqALT  4547  nfpw  4551  pw0  4742  pwpw0  4743  pwsn  4828  pwsnOLD  4829  vpwex  5295  abssexg  5300  orduniss2  7655  mapex  8579  ssenen  8887  domtriomlem  10129  npex  10673  ustval  23262  avril1  28728  fineqvpow  32965  dfon2lem2  33666