Definition df-pw 4577

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 30356). We will later introduce the Axiom of Power Sets ax-pow 5335, which can be expressed in class notation per pwexg 5348. Still later we will prove, in hashpw 14452, 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 4575	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1539	. . . 4 class 𝑥
5	4, 1	wss 3926	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2713	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1540	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4578  pweqALT  4590  nfpw  4594  pw0  4788  pwpw0  4789  pwsn  4876  vpwex  5347  abssexg  5352  orduniss2  7825  mapexOLD  8844  ssenen  9163  domtriomlem  10454  npex  10998  ustval  24139  avril1  30390  fineqvpow  35073  dfon2lem2  35748  bj-velpwALT  37017