Definition df-pw 4597

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 30176). We will later introduce the Axiom of Power Sets ax-pow 5354, which can be expressed in class notation per pwexg 5367. Still later we will prove, in hashpw 14397, 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 4595	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1532	. . . 4 class 𝑥
5	4, 1	wss 3941	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2701	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1533	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4598  pweqALT  4610  nfpw  4614  pw0  4808  pwpw0  4809  pwsn  4893  vpwex  5366  abssexg  5371  orduniss2  7815  mapex  8823  ssenen  9148  domtriomlem  10434  npex  10978  ustval  24051  avril1  30210  fineqvpow  34614  dfon2lem2  35278  bj-velpwALT  36434