Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-om Structured version   Unicode version

Definition df-om 4849
 Description: Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e. all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 4850 for an alternate definition. Later, when we assume the Axiom of Infinity, we show is a set in omex 7601, and can then be defined per dfom3 7605 (the smallest inductive set) and dfom4 7607. Note: the natural numbers are a subset of the ordinal numbers df-on 4588. They are completely different from the natural numbers (df-nn 10006) that are a subset of the complex numbers defined much later in our development, although the two sets have analogous properties and operations defined on them. (Contributed by NM, 15-May-1994.)
Assertion
Ref Expression
df-om
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-om
StepHypRef Expression
1 com 4848 . 2
2 vy . . . . . . 7
32cv 1652 . . . . . 6
43wlim 4585 . . . . 5
5 vx . . . . . 6
65, 2wel 1727 . . . . 5
74, 6wi 4 . . . 4
87, 2wal 1550 . . 3
9 con0 4584 . . 3
108, 5, 9crab 2711 . 2
111, 10wceq 1653 1
 Colors of variables: wff set class This definition is referenced by:  dfom2  4850  elom  4851
 Copyright terms: Public domain W3C validator