Definition df-pw 4381

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 27845). We will later introduce the Axiom of Power Sets ax-pow 5066, which can be expressed in class notation per pwexg 5079. Still later we will prove, in hashpw 13513, 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 4379	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1657	. . . 4 class 𝑥
5	4, 1	wss 3799	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2812	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1658	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  pweq  4382  elpw  4385  nfpw  4393  pw0  4562  pwpw0  4563  pwsn  4651  pwsnALT  4652  vpwex  5078  abssexg  5082  orduniss2  7295  mapex  8129  ssenen  8404  domtriomlem  9580  npex  10124  ustval  22377  avril1  27878  dfon2lem2  32228