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Definition df-r1 8510
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 8537). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 8514, r1suc 8516, and r1lim 8518. Theorem r1val1 8532 shows a recursive definition that works for all values, and theorems r1val2 8583 and r1val3 8584 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 8508 . 2 class 𝑅1
2 vx . . . 4 setvar 𝑥
3 cvv 3173 . . . 4 class V
42cv 1474 . . . . 5 class 𝑥
54cpw 4108 . . . 4 class 𝒫 𝑥
62, 3, 5cmpt 4643 . . 3 class (𝑥 ∈ V ↦ 𝒫 𝑥)
7 c0 3874 . . 3 class
86, 7crdg 7392 . 2 class rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
91, 8wceq 1475 1 wff 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
Colors of variables: wff setvar class
This definition is referenced by:  r1funlim  8512  r1fnon  8513  r10  8514  r1sucg  8515  r1limg  8517
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