Definition df-pw 4533

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 30487). We will later introduce the Axiom of Power Sets ax-pow 5296, which can be expressed in class notation per pwexg 5309. Still later we will prove, in hashpw 14387, 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 4531	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1541	. . . 4 class 𝑥
5	4, 1	wss 3885	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2713	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1542	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4534  pweqALT  4546  nfpw  4550  pw0  4745  pwpw0  4746  pwsn  4833  vpwex  5308  abssexg  5313  orduniss2  7773  mapexOLD  8768  ssenen  9078  domtriomlem  10353  npex  10898  ustval  24156  avril1  30521  fineqvpow  35247  dfon2lem2  35952  bj-velpwALT  37348