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| 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 12268. (Contributed by Scott Fenton, 17-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| df-n0s | ⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cnn0s 28319 | . 2 class ℕ0s | |
| 2 | vx | . . . . 5 setvar 𝑥 | |
| 3 | cvv 3479 | . . . . 5 class V | |
| 4 | 2 | cv 1538 | . . . . . 6 class 𝑥 | 
| 5 | c1s 27869 | . . . . . 6 class 1s | |
| 6 | cadds 27993 | . . . . . 6 class +s | |
| 7 | 4, 5, 6 | co 7432 | . . . . 5 class (𝑥 +s 1s ) | 
| 8 | 2, 3, 7 | cmpt 5224 | . . . 4 class (𝑥 ∈ V ↦ (𝑥 +s 1s )) | 
| 9 | c0s 27868 | . . . 4 class 0s | |
| 10 | 8, 9 | crdg 8450 | . . 3 class rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) | 
| 11 | com 7888 | . . 3 class ω | |
| 12 | 10, 11 | cima 5687 | . 2 class (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω) | 
| 13 | 1, 12 | wceq 1539 | 1 wff ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω) | 
| Colors of variables: wff setvar class | 
| This definition is referenced by: n0sex 28323 peano5n0s 28325 n0ssno 28326 0n0s 28335 peano2n0s 28336 n0sind 28338 seqn0sfn 28358 | 
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