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Definition df-r1 8612
 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 8639). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 8616, r1suc 8618, and r1lim 8620. Theorem r1val1 8634 shows a recursive definition that works for all values, and theorems r1val2 8685 and r1val3 8686 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 ∈ V ↦ 𝒫 𝑥), ∅)

Detailed syntax breakdown of Definition df-r1
StepHypRef Expression
1 cr1 8610 . 2 class 𝑅1
2 vx . . . 4 setvar 𝑥
3 cvv 3195 . . . 4 class V
42cv 1480 . . . . 5 class 𝑥
54cpw 4149 . . . 4 class 𝒫 𝑥
62, 3, 5cmpt 4720 . . 3 class (𝑥 ∈ V ↦ 𝒫 𝑥)
7 c0 3907 . . 3 class
86, 7crdg 7490 . 2 class rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
91, 8wceq 1481 1 wff 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
 Colors of variables: wff setvar class This definition is referenced by:  r1funlim  8614  r1fnon  8615  r10  8616  r1sucg  8617  r1limg  8619
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