Theorem nnn0s 28277

Description: A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)

Assertion

Ref	Expression
nnn0s	⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)

Proof of Theorem nnn0s

Step	Hyp	Ref	Expression
1		nnssn0s 28271	. 2 ⊢ ℕs ⊆ ℕ0s
2	1	sseli 3959	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ∈ wcel 2109  ℕ_0scnn0s 28263  ℕ_scnns 28264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-dif 3934  df-ss 3948  df-nns 28266

This theorem is referenced by:  elzn0s  28343  eln0zs  28345  zseo  28365  addhalfcut  28391  zs12ge0  28399