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 30963). We will later introduce the Axiom of Power Sets ax-pow 5326, which can be expressed in class notation per pwexg 5339. Still later we will prove, in hashpw 14548, 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 1569	. . . 4 class 𝑥
5	4, 1	wss 3898	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2738	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1570	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is used by:  elpwg  4559  pweqALT  4571  nfpw  4575  pw0  4772  pwpw0  4773  pwsn  4859  vpwex  5338  abssexg  5343  orduniss2  7827  ssenen  9148  domtriomlem  10491  npex  11042  ustval  24483  avril1  30997  fineqvpow  35708  dfon2lem2  36468  bj-velpwALT  37888