Definition df-pw 4557

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 30508). We will later introduce the Axiom of Power Sets ax-pow 5311, which can be expressed in class notation per pwexg 5324. Still later we will prove, in hashpw 14363, 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 4555	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1541	. . . 4 class 𝑥
5	4, 1	wss 3902	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2715	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1542	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4558  pweqALT  4570  nfpw  4574  pw0  4769  pwpw0  4770  pwsn  4857  vpwex  5323  abssexg  5328  orduniss2  7777  mapexOLD  8773  ssenen  9083  domtriomlem  10356  npex  10901  ustval  24151  avril1  30542  fineqvpow  35273  dfon2lem2  35978  bj-velpwALT  37256