MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-r1 Unicode version

Definition df-r1 8203
Description: Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation ( ). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 8230). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 8207, r1suc 8209, and r1lim 8211. Theorem r1val1 8225 shows a recursive definition that works for all values, and theorems r1val2 8276 and r1val3 8277 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477), with a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.)
Assertion
Ref Expression
df-r1

Detailed syntax breakdown of Definition df-r1
StepHypRef Expression
1 cr1 8201 . 2
2 vx . . . 4
3 cvv 3109 . . . 4
42cv 1394 . . . . 5
54cpw 4012 . . . 4
62, 3, 5cmpt 4510 . . 3
7 c0 3784 . . 3
86, 7crdg 7094 . 2
91, 8wceq 1395 1
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  8205  r1fnon  8206  r10  8207  r1sucg  8208  r1limg  8210
  Copyright terms: Public domain W3C validator