Theorem nnn0s 28718

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

Assertion

Ref	Expression
nnn0s	⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)

Proof of Theorem nnn0s

Step	Hyp	Ref	Expression
1		nnssn0s 28712	. 2 ⊢ ℕs ⊆ ℕ0s
2	1	sseli 3904	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ℕ_0scn0s 28703  ℕ_scnns 28704

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-dif 4047  df-nns 28706

This theorem is used by:  elzn0s  28789  eln0zs  28791  zseo  28813  addhalfcut  28850  bdaypw2n0bndlem  28854  bdayfinbndlem1  28858  z12bdaylem2  28862  z12sge0  28874