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Definition df-pw 3332
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 is { 3 , 5 , 7 }, then 𝒫 A is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove 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 𝒫 A = {xxA}
Distinct variable group:   x,A

Detailed syntax breakdown of Definition df-pw
StepHypRef Expression
1 cA . . 3 class A
21cpw 3330 . 2 class 𝒫 A
3 vx . . . . 5 setvar x
43cv 1225 . . . 4 class x
54, 1wss 2890 . . 3 wff xA
65, 3cab 2004 . 2 class {xxA}
72, 6wceq 1226 1 wff 𝒫 A = {xxA}
Colors of variables: wff set class
This definition is referenced by:  pweq  3333  elpw  3336  nfpw  3342  pwss  3345  pw0  3481  snsspw  3505  pwsnss  3544  pwex  3902  abssexg  3904  iunpw  4157  iotass  4807  bdcpw  7235
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