Definition df-pw 3725

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 ℘A = {∅, { 3 }, { 5 }, { 7 }, { 3 , 5 }, { 3 , 7 }, { 5 , 7 }, { 3 , 5 , 7 }} (ex-pw in set.mm). We will later introduce the Axiom of Power Sets ax-pow in set.mm, which can be expressed in class notation per pwexg 4329. Still later we will prove, in hashpw in set.mm, 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, 5-Aug-1993.)

Assertion

Ref	Expression
df-pw	⊢ ℘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 3723	. 2 class ℘A
3		vx	. . . . 5 setvar x
4	3	cv 1641	. . . 4 class x
5	4, 1	wss 3258	. . 3 wff x ⊆ A
6	5, 3	cab 2339	. 2 class {x ∣ x ⊆ A}
7	2, 6	wceq 1642	1 wff ℘A = {x ∣ x ⊆ A}

This definition is referenced by:  pweq  3726  elpw  3729  nfpw  3734  pwss  3737  pwpw0  3856  snsspw  3878  pwsn  3882  pwsnALT  3883  pw0  4161  eqpwrelk  4479  srelk  4525  pmex  6006  pmvalg  6011  enpw1pw  6076