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Definition df-iom 4230
Description: Define the class of natural numbers as the smallest inductive set, which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82.

Note: the natural numbers 𝜔 are a subset of the ordinal numbers df-on 4044. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers with analogous properties and operations, but they will be different sets. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4231 instead for naming consistency with set.mm. (New usage is discouraged.)

Assertion
Ref Expression
df-iom 𝜔 = {x ∣ (∅ x y x suc y x)}
Distinct variable group:   x,y

Detailed syntax breakdown of Definition df-iom
StepHypRef Expression
1 com 4229 . 2 class 𝜔
2 c0 3193 . . . . . 6 class
3 vx . . . . . . 7 setvar x
43cv 1223 . . . . . 6 class x
52, 4wcel 1367 . . . . 5 wff x
6 vy . . . . . . . . 9 setvar y
76cv 1223 . . . . . . . 8 class y
87csuc 4041 . . . . . . 7 class suc y
98, 4wcel 1367 . . . . . 6 wff suc y x
109, 6, 4wral 2276 . . . . 5 wff y x suc y x
115, 10wa 97 . . . 4 wff (∅ x y x suc y x)
1211, 3cab 2000 . . 3 class {x ∣ (∅ x y x suc y x)}
1312cint 3579 . 2 class {x ∣ (∅ x y x suc y x)}
141, 13wceq 1224 1 wff 𝜔 = {x ∣ (∅ x y x suc y x)}
Colors of variables: wff set class
This definition is referenced by:  dfom3  4231
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