Theorem nnn0sd 28257

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

Hypothesis

Ref	Expression
nnn0sd.1	⊢ (𝜑 → 𝐴 ∈ ℕs)

Assertion

Ref	Expression
nnn0sd	⊢ (𝜑 → 𝐴 ∈ ℕ0s)

Proof of Theorem nnn0sd

Step	Hyp	Ref	Expression
1		nnssn0s 28250	. 2 ⊢ ℕs ⊆ ℕ0s
2		nnn0sd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕs)
3	1, 2	sselid 3927	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ∈ wcel 2111  ℕ_0scnn0s 28242  ℕ_scnns 28243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-ss 3914  df-nns 28245

This theorem is referenced by:  eucliddivs  28301