Definition df-pw 4551

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 30411). We will later introduce the Axiom of Power Sets ax-pow 5305, which can be expressed in class notation per pwexg 5318. Still later we will prove, in hashpw 14345, 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 4549	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1540	. . . 4 class 𝑥
5	4, 1	wss 3898	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2711	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1541	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4552  pweqALT  4564  nfpw  4568  pw0  4763  pwpw0  4764  pwsn  4851  vpwex  5317  abssexg  5322  orduniss2  7769  mapexOLD  8762  ssenen  9071  domtriomlem  10340  npex  10884  ustval  24119  avril1  30445  fineqvpow  35159  dfon2lem2  35847  bj-velpwALT  37118