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Definition df-r1 9765
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 9792). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as Theorems r10 9769, r1suc 9771, and r1lim 9773. Theorem r1val1 9787 shows a recursive definition that works for all values, and Theorems r1val2 9838 and r1val3 9839 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 9763 . 2 class 𝑅1
2 vx . . . 4 setvar 𝑥
3 cvv 3473 . . . 4 class V
42cv 1539 . . . . 5 class 𝑥
54cpw 4602 . . . 4 class 𝒫 𝑥
62, 3, 5cmpt 5231 . . 3 class (𝑥 ∈ V ↦ 𝒫 𝑥)
7 c0 4322 . . 3 class
86, 7crdg 8415 . 2 class rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
91, 8wceq 1540 1 wff 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  9767  r1fnon  9768  r10  9769  r1sucg  9770  r1limg  9772
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