Definition df-pw 4544

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 30496). We will later introduce the Axiom of Power Sets ax-pow 5306, which can be expressed in class notation per pwexg 5319. Still later we will prove, in hashpw 14395, 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 4542	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1541	. . . 4 class 𝑥
5	4, 1	wss 3890	. . 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  4545  pweqALT  4557  nfpw  4561  pw0  4756  pwpw0  4757  pwsn  4844  vpwex  5318  abssexg  5323  orduniss2  7781  mapexOLD  8776  ssenen  9086  domtriomlem  10361  npex  10906  ustval  24165  avril1  30530  fineqvpow  35256  dfon2lem2  35961  bj-velpwALT  37357