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

Definition df-r1 7967
Description: Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation ( R1). 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 7994). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 7971, r1suc 7973, and r1lim 7975. Theorem r1val1 7989 shows a recursive definition that works for all values, and theorems r1val2 8040 and r1val3 8041 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),  _V 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  |-  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )

Detailed syntax breakdown of Definition df-r1
StepHypRef Expression
1 cr1 7965 . 2  class  R1
2 vx . . . 4  setvar  x
3 cvv 2970 . . . 4  class  _V
42cv 1363 . . . . 5  class  x
54cpw 3857 . . . 4  class  ~P x
62, 3, 5cmpt 4347 . . 3  class  ( x  e.  _V  |->  ~P x
)
7 c0 3634 . . 3  class  (/)
86, 7crdg 6861 . 2  class  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )
91, 8wceq 1364 1  wff  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  7969  r1fnon  7970  r10  7971  r1sucg  7972  r1limg  7974
  Copyright terms: Public domain W3C validator