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Definition df-r1 8235
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 8262). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 8239, r1suc 8241, and r1lim 8243. Theorem r1val1 8257 shows a recursive definition that works for all values, and theorems r1val2 8308 and r1val3 8309 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 8233 . 2  class  R1
2 vx . . . 4  setvar  x
3 cvv 3045 . . . 4  class  _V
42cv 1443 . . . . 5  class  x
54cpw 3951 . . . 4  class  ~P x
62, 3, 5cmpt 4461 . . 3  class  ( x  e.  _V  |->  ~P x
)
7 c0 3731 . . 3  class  (/)
86, 7crdg 7127 . 2  class  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )
91, 8wceq 1444 1  wff  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  8237  r1fnon  8238  r10  8239  r1sucg  8240  r1limg  8242
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