Theorem nnn0s 28695

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 28689	. 2 ⊢ ℕs ⊆ ℕ0s
2	1	sseli 3927	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ℕ_0scn0s 28680  ℕ_scnns 28681

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 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-ss 3916  df-nns 28683

This theorem is used by:  elzn0s  28766  eln0zs  28768  zseo  28790  addhalfcut  28827  bdaypw2n0bndlem  28831  bdayfinbndlem1  28835  z12bdaylem2  28839  z12sge0  28851