Definition df-pw 4559

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 30633). We will later introduce the Axiom of Power Sets ax-pow 5324, which can be expressed in class notation per pwexg 5337. Still later we will prove, in hashpw 14451, 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 4557	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1561	. . . 4 class 𝑥
5	4, 1	wss 3906	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2742	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1562	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  elpwg  4560  pweqALT  4572  nfpw  4576  pw0  4772  pwpw0  4773  pwsn  4860  vpwex  5336  abssexg  5341  orduniss2  7815  ssenen  9125  domtriomlem  10401  npex  10946  ustval  24265  avril1  30667  fineqvpow  35415  dfon2lem2  36137  bj-velpwALT  37543