ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-pw GIF version

Definition df-pw 3459
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 3457 . 2 class 𝒫 𝐴
3 vx . . . . 5 setvar 𝑥
43cv 1298 . . . 4 class 𝑥
54, 1wss 3021 . . 3 wff 𝑥𝐴
65, 3cab 2086 . 2 class {𝑥𝑥𝐴}
72, 6wceq 1299 1 wff 𝒫 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff set class
This definition is referenced by:  pweq  3460  elpw  3463  nfpw  3470  pwss  3473  pw0  3614  snsspw  3638  pwsnss  3677  vpwex  4043  abssexg  4046  iunpw  4339  iotass  5041  mapex  6478  ssenen  6674  tgvalex  12001  bdcpw  12648
  Copyright terms: Public domain W3C validator