Definition df-pw 4533

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 30519). We will later introduce the Axiom of Power Sets ax-pow 5296, which can be expressed in class notation per pwexg 5309. Still later we will prove, in hashpw 14393, 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 4531	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1547	. . . 4 class 𝑥
5	4, 1	wss 3884	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2719	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1548	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4534  pweqALT  4546  nfpw  4550  pw0  4745  pwpw0  4746  pwsn  4833  vpwex  5308  abssexg  5313  orduniss2  7776  mapexOLD  8773  ssenen  9083  domtriomlem  10360  npex  10905  ustval  24189  avril1  30553  fineqvpow  35309  dfon2lem2  36023  bj-velpwALT  37419