Theorem nnn0sd 28334

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 28327	. 2 ⊢ ℕs ⊆ ℕ0s
2		nnn0sd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℕs)
3	1, 2	sselid 3980	1 ⊢ (𝜑 → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ∈ wcel 2107  ℕ_0scnn0s 28319  ℕ_scnns 28320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3953  df-ss 3967  df-nns 28322

This theorem is referenced by: (None)