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Definition df-pw 3358
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 3356 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1242 . . . 4 class 𝑥
54, 1wss 2914 . . 3 wff 𝑥𝐴
65, 3cab 2026 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1243 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff set class
This definition is referenced by:  pweq  3359  elpw  3362  nfpw  3368  pwss  3371  pw0  3508  snsspw  3532  pwsnss  3571  pwex  3929  abssexg  3931  iunpw  4198  iotass  4862  bdcpw  9853
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