Theorem nnn0sd 28647

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 28640	. 2 ⊢ ℕs ⊆ ℕ0s
2		nnn0sd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕs)
3	1, 2	sselid 3928	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0s)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ℕ_0scn0s 28631  ℕ_scnns 28632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 3901  df-ss 3915  df-nns 28634

This theorem is used by:  eucliddivs  28695