Theorem nnn0s 28478

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 28472	. 2 ⊢ ℕs ⊆ ℕ0s
2	1	sseli 3935	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ∈ wcel 2145  ℕ_0scn0s 28463  ℕ_scnns 28464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3910  df-ss 3924  df-nns 28466

This theorem is referenced by:  elzn0s  28549  eln0zs  28551  zseo  28573  addhalfcut  28610  bdaypw2n0bndlem  28614  bdayfinbndlem1  28618  z12bdaylem2  28622  z12sge0  28634