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Definition df-r1 8182
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 8209). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 8186, r1suc 8188, and r1lim 8190. Theorem r1val1 8204 shows a recursive definition that works for all values, and theorems r1val2 8255 and r1val3 8256 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 8180 . 2  class  R1
2 vx . . . 4  setvar  x
3 cvv 3113 . . . 4  class  _V
42cv 1378 . . . . 5  class  x
54cpw 4010 . . . 4  class  ~P x
62, 3, 5cmpt 4505 . . 3  class  ( x  e.  _V  |->  ~P x
)
7 c0 3785 . . 3  class  (/)
86, 7crdg 7075 . 2  class  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )
91, 8wceq 1379 1  wff  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  8184  r1fnon  8185  r10  8186  r1sucg  8187  r1limg  8189
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