Theorem nnne0s 28567

Description: A surreal positive integer is nonzero. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
nnne0s	⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

Proof of Theorem nnne0s

Step	Hyp	Ref	Expression
1		eldifsni 4763	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s )
2		df-nns 28545	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2884	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ≠ wne 2961   ∖ cdif 3905  {csn 4594   0_s c0s 28035  ℕ_0scn0s 28542  ℕ_scnns 28543

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 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-v 3460  df-dif 3911  df-sn 4595  df-nns 28545

This theorem is used by:  nnsgt0  28569  2ne0s  28650  expnnsval  28656  recut  28724  elreno2  28725  renegscl  28728  readdscl  28729  remulscllem1  28730  remulscl  28732