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Mirrors > Home > ILE Home > Th. List > df-iom | Unicode 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 are a subset of the ordinal numbers df-on 4340. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8849) with analogous properties and operations, but they will be different sets. We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7141. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4563 instead for naming consistency with set.mm. (New usage is discouraged.) |
Ref | Expression |
---|---|
df-iom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | com 4561 | . 2 | |
2 | c0 3404 | . . . . . 6 | |
3 | vx | . . . . . . 7 | |
4 | 3 | cv 1341 | . . . . . 6 |
5 | 2, 4 | wcel 2135 | . . . . 5 |
6 | vy | . . . . . . . . 9 | |
7 | 6 | cv 1341 | . . . . . . . 8 |
8 | 7 | csuc 4337 | . . . . . . 7 |
9 | 8, 4 | wcel 2135 | . . . . . 6 |
10 | 9, 6, 4 | wral 2442 | . . . . 5 |
11 | 5, 10 | wa 103 | . . . 4 |
12 | 11, 3 | cab 2150 | . . 3 |
13 | 12 | cint 3818 | . 2 |
14 | 1, 13 | wceq 1342 | 1 |
Colors of variables: wff set class |
This definition is referenced by: dfom3 4563 |
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