Definition df-pw 4555

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 30391). We will later introduce the Axiom of Power Sets ax-pow 5307, which can be expressed in class notation per pwexg 5320. Still later we will prove, in hashpw 14361, 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 4553	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1539	. . . 4 class 𝑥
5	4, 1	wss 3905	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2707	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1540	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4556  pweqALT  4568  nfpw  4572  pw0  4766  pwpw0  4767  pwsn  4854  vpwex  5319  abssexg  5324  orduniss2  7772  mapexOLD  8766  ssenen  9075  domtriomlem  10355  npex  10899  ustval  24106  avril1  30425  fineqvpow  35070  dfon2lem2  35757  bj-velpwALT  37026