Definition df-pw 4558

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 30522). We will later introduce the Axiom of Power Sets ax-pow 5314, which can be expressed in class notation per pwexg 5327. Still later we will prove, in hashpw 14373, 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 4556	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1541	. . . 4 class 𝑥
5	4, 1	wss 3903	. . 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  4559  pweqALT  4571  nfpw  4575  pw0  4770  pwpw0  4771  pwsn  4858  vpwex  5326  abssexg  5331  orduniss2  7787  mapexOLD  8783  ssenen  9093  domtriomlem  10366  npex  10911  ustval  24164  avril1  30556  fineqvpow  35299  dfon2lem2  36004  bj-velpwALT  37328