Definition df-n0s 28215

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 12194. (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		cnn0s 28213	. 2 class ℕ0s
2		vx	. . . . 5 setvar 𝑥
3		cvv 3450	. . . . 5 class V
4	2	cv 1539	. . . . . 6 class 𝑥
5		c1s 27742	. . . . . 6 class 1s
6		cadds 27873	. . . . . 6 class +s
7	4, 5, 6	co 7390	. . . . 5 class (𝑥 +s 1s )
8	2, 3, 7	cmpt 5191	. . . 4 class (𝑥 ∈ V ↦ (𝑥 +s 1s ))
9		c0s 27741	. . . 4 class 0s
10	8, 9	crdg 8380	. . 3 class rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s )
11		com 7845	. . 3 class ω
12	10, 11	cima 5644	. 2 class (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
13	1, 12	wceq 1540	1 wff ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)

This definition is referenced by:  n0sex  28217  peano5n0s  28219  n0ssno  28220  0n0s  28229  peano2n0s  28230  n0sind  28232  seqn0sfn  28257