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Definition df-r1 7971
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 7998). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 7975, r1suc 7977, and r1lim 7979. Theorem r1val1 7993 shows a recursive definition that works for all values, and theorems r1val2 8044 and r1val3 8045 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 7969 . 2  class  R1
2 vx . . . 4  setvar  x
3 cvv 2972 . . . 4  class  _V
42cv 1368 . . . . 5  class  x
54cpw 3860 . . . 4  class  ~P x
62, 3, 5cmpt 4350 . . 3  class  ( x  e.  _V  |->  ~P x
)
7 c0 3637 . . 3  class  (/)
86, 7crdg 6865 . 2  class  rec (
( x  e.  _V  |->  ~P x ) ,  (/) )
91, 8wceq 1369 1  wff  R1  =  rec ( ( x  e. 
_V  |->  ~P x ) ,  (/) )
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  7973  r1fnon  7974  r10  7975  r1sucg  7976  r1limg  7978
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