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Definition df-r1 8877
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 8904). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 8881, r1suc 8883, and r1lim 8885. Theorem r1val1 8899 shows a recursive definition that works for all values, and theorems r1val2 8950 and r1val3 8951 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 8875 . 2 class 𝑅1
2 vx . . . 4 setvar 𝑥
3 cvv 3398 . . . 4 class V
42cv 1636 . . . . 5 class 𝑥
54cpw 4358 . . . 4 class 𝒫 𝑥
62, 3, 5cmpt 4930 . . 3 class (𝑥 ∈ V ↦ 𝒫 𝑥)
7 c0 4123 . . 3 class
86, 7crdg 7744 . 2 class rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
91, 8wceq 1637 1 wff 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  8879  r1fnon  8880  r10  8881  r1sucg  8882  r1limg  8884
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