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 30791). We will later introduce the Axiom of Power Sets ax-pow 5335, which can be expressed in class notation per pwexg 5348. Still later we will prove, in hashpw 14480, 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 1568	. . . 4 class 𝑥
5	4, 1	wss 3904	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2740	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1569	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is used by:  elpwg  4564  pweqALT  4576  nfpw  4580  pw0  4777  pwpw0  4778  pwsn  4864  vpwex  5347  abssexg  5352  orduniss2  7827  ssenen  9137  domtriomlem  10432  npex  10977  ustval  24371  avril1  30825  fineqvpow  35536  dfon2lem2  36282  bj-velpwALT  37717