Theorem nnn0sd 28348

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 28341	. 2 ⊢ ℕs ⊆ ℕ0s
2		nnn0sd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕs)
3	1, 2	sselid 3993	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ∈ wcel 2106  ℕ_0scnn0s 28333  ℕ_scnns 28334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-ss 3980  df-nns 28336

This theorem is referenced by: (None)