Definition df-n0s 28465

Description: Define the set of non-negative surreal integers. This set behaves similarly to ω and ℕ₀, but it is a set of surreal numbers. Like those two sets, it satisfies the Peano axioms and is closed under (surreal) addition and multiplication. Compare df-nn 12225. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
df-n0s	⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)

Detailed syntax breakdown of Definition df-n0s

Step	Hyp	Ref	Expression
1		cn0s 28463	. 2 class ℕ0s
2		vx	. . . . 5 setvar 𝑥
3		cvv 3457	. . . . 5 class V
4	2	cv 1562	. . . . . 6 class 𝑥
5		c1s 27957	. . . . . 6 class 1s
6		cadds 28110	. . . . . 6 class +s
7	4, 5, 6	co 7400	. . . . 5 class (𝑥 +s 1s )
8	2, 3, 7	cmpt 5186	. . . 4 class (𝑥 ∈ V ↦ (𝑥 +s 1s ))
9		c0s 27956	. . . 4 class 0s
10	8, 9	crdg 8384	. . 3 class rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s )
11		com 7850	. . 3 class ω
12	10, 11	cima 5655	. 2 class (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
13	1, 12	wceq 1563	1 wff ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)

This definition is referenced by:  n0sexg  28467  peano5n0s  28470  n0ssno  28471  0n0s  28480  peano2n0s  28481  n0sind  28484  seqn0sfn  28511