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Mirrors > Home > ILE Home > Th. List > df-iom | GIF version |
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 4071. 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 4258 instead for naming consistency with set.mm. (New usage is discouraged.) |
Ref | Expression |
---|---|
df-iom | ⊢ 𝜔 = ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | com 4256 | . 2 class 𝜔 | |
2 | c0 3218 | . . . . . 6 class ∅ | |
3 | vx | . . . . . . 7 setvar x | |
4 | 3 | cv 1241 | . . . . . 6 class x |
5 | 2, 4 | wcel 1390 | . . . . 5 wff ∅ ∈ x |
6 | vy | . . . . . . . . 9 setvar y | |
7 | 6 | cv 1241 | . . . . . . . 8 class y |
8 | 7 | csuc 4068 | . . . . . . 7 class suc y |
9 | 8, 4 | wcel 1390 | . . . . . 6 wff suc y ∈ x |
10 | 9, 6, 4 | wral 2300 | . . . . 5 wff ∀y ∈ x suc y ∈ x |
11 | 5, 10 | wa 97 | . . . 4 wff (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x) |
12 | 11, 3 | cab 2023 | . . 3 class {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)} |
13 | 12 | cint 3606 | . 2 class ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)} |
14 | 1, 13 | wceq 1242 | 1 wff 𝜔 = ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)} |
Colors of variables: wff set class |
This definition is referenced by: dfom3 4258 |
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