Definition df-pw 4606

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 30457). We will later introduce the Axiom of Power Sets ax-pow 5370, which can be expressed in class notation per pwexg 5383. Still later we will prove, in hashpw 14471, 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 4604	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1535	. . . 4 class 𝑥
5	4, 1	wss 3962	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2711	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1536	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4607  pweqALT  4619  nfpw  4623  pw0  4816  pwpw0  4817  pwsn  4904  vpwex  5382  abssexg  5387  orduniss2  7852  mapexOLD  8870  ssenen  9189  domtriomlem  10479  npex  11023  ustval  24226  avril1  30491  fineqvpow  35088  dfon2lem2  35765  bj-velpwALT  37035