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Definition df-pw 3724
 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 4328. 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
StepHypRef Expression
1 cA . . 3 class A
21cpw 3722 . 2 class A
3 vx . . . . 5 setvar x
43cv 1641 . . . 4 class x
54, 1wss 3257 . . 3 wff x A
65, 3cab 2339 . 2 class {x x A}
72, 6wceq 1642 1 wff A = {x x A}
 Colors of variables: wff setvar class This definition is referenced by:  pweq  3725  elpw  3728  nfpw  3733  pwss  3736  pwpw0  3855  snsspw  3877  pwsn  3881  pwsnALT  3882  pw0  4160  eqpwrelk  4478  srelk  4524  pmex  6005  pmvalg  6010  enpw1pw  6075
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