Definition df-pw 4379

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 27843). We will later introduce the Axiom of Power Sets ax-pow 5064, which can be expressed in class notation per pwexg 5077. Still later we will prove, in hashpw 13511, 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 4377	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1657	. . . 4 class 𝑥
5	4, 1	wss 3797	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2810	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1658	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  pweq  4380  elpw  4383  nfpw  4391  pw0  4560  pwpw0  4561  pwsn  4649  pwsnALT  4650  vpwex  5076  abssexg  5080  orduniss2  7293  mapex  8127  ssenen  8402  domtriomlem  9578  npex  10122  ustval  22375  avril1  27876  dfon2lem2  32226