Definition df-pw 4563

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 30746). We will later introduce the Axiom of Power Sets ax-pow 5336, which can be expressed in class notation per pwexg 5349. Still later we will prove, in hashpw 14472, 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 4561	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1567	. . . 4 class 𝑥
5	4, 1	wss 3904	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2739	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1568	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4564  pweqALT  4576  nfpw  4580  pw0  4777  pwpw0  4778  pwsn  4864  vpwex  5348  abssexg  5353  orduniss2  7828  ssenen  9138  domtriomlem  10425  npex  10970  ustval  24339  avril1  30780  fineqvpow  35482  dfon2lem2  36228  bj-velpwALT  37633