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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 4260. 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 4476 instead for naming consistency with set.mm. (New usage is discouraged.) |
Ref | Expression |
---|---|
df-iom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | com 4474 | . 2 | |
2 | c0 3333 | . . . . . 6 | |
3 | vx | . . . . . . 7 | |
4 | 3 | cv 1315 | . . . . . 6 |
5 | 2, 4 | wcel 1465 | . . . . 5 |
6 | vy | . . . . . . . . 9 | |
7 | 6 | cv 1315 | . . . . . . . 8 |
8 | 7 | csuc 4257 | . . . . . . 7 |
9 | 8, 4 | wcel 1465 | . . . . . 6 |
10 | 9, 6, 4 | wral 2393 | . . . . 5 |
11 | 5, 10 | wa 103 | . . . 4 |
12 | 11, 3 | cab 2103 | . . 3 |
13 | 12 | cint 3741 | . 2 |
14 | 1, 13 | wceq 1316 | 1 |
Colors of variables: wff set class |
This definition is referenced by: dfom3 4476 |
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