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Definition df-r1 8180
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 8207). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 8184, r1suc 8186, and r1lim 8188. Theorem r1val1 8202 shows a recursive definition that works for all values, and theorems r1val2 8253 and r1val3 8254 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 8178 . 2  class  R1
2 vx . . . 4  setvar  x
3 cvv 3093 . . . 4  class  _V
42cv 1380 . . . . 5  class  x
54cpw 3993 . . . 4  class  ~P x
62, 3, 5cmpt 4491 . . 3  class  ( x  e.  _V  |->  ~P x
)
7 c0 3767 . . 3  class  (/)
86, 7crdg 7073 . 2  class  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )
91, 8wceq 1381 1  wff  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  8182  r1fnon  8183  r10  8184  r1sucg  8185  r1limg  8187
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