Definition df-pw 4160

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 27286). We will later introduce the Axiom of Power Sets ax-pow 4843, which can be expressed in class notation per pwexg 4850. Still later we will prove, in hashpw 13223, 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 4158	. 2 class 𝒫 𝐴
3		vx	. . . . 5 setvar 𝑥
4	3	cv 1482	. . . 4 class 𝑥
5	4, 1	wss 3574	. . 3 wff 𝑥 ⊆ 𝐴
6	5, 3	cab 2608	. 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	2, 6	wceq 1483	1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}

This definition is referenced by:  pweq  4161  elpw  4164  nfpw  4172  pw0  4343  pwpw0  4344  pwsn  4428  pwsnALT  4429  pwex  4848  abssexg  4851  orduniss2  7033  mapex  7863  ssenen  8134  domtriomlem  9264  npex  9808  ustval  22006  avril1  27319  dfon2lem2  31689