Definition df-pw 4604

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 29672). We will later introduce the Axiom of Power Sets ax-pow 5363, which can be expressed in class notation per pwexg 5376. 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 4602	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1541	. . . 4 class 𝑥
5	4, 1	wss 3948	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2710	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1542	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4605  pweqALT  4617  nfpw  4621  pw0  4815  pwpw0  4816  pwsn  4900  pwsnOLD  4901  vpwex  5375  abssexg  5380  orduniss2  7818  mapex  8823  ssenen  9148  domtriomlem  10434  npex  10978  ustval  23699  avril1  29706  fineqvpow  34085  dfon2lem2  34745  bj-velpwALT  35923