Theorem nnn0sd 28497

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 28490	. 2 ⊢ ℕs ⊆ ℕ0s
2		nnn0sd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕs)
3	1, 2	sselid 3934	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ∈ wcel 2141  ℕ_0scn0s 28481  ℕ_scnns 28482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3907  df-ss 3921  df-nns 28484

This theorem is referenced by:  eucliddivs  28545