Theorem nnn0s 28498

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 28492	. 2 ⊢ ℕs ⊆ ℕ0s
2	1	sseli 3934	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ∈ wcel 2143  ℕ_0scn0s 28483  ℕ_scnns 28484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3909  df-ss 3923  df-nns 28486

This theorem is referenced by:  elzn0s  28569  eln0zs  28571  zseo  28593  addhalfcut  28630  bdaypw2n0bndlem  28634  bdayfinbndlem1  28638  z12bdaylem2  28642  z12sge0  28654