Theorem nnne0s 28729

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 4749	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s )
2		df-nns 28707	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2878	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ≠ wne 2955   ∖ cdif 4045  {csn 4584   0_s c0s 28197  ℕ_0scn0s 28704  ℕ_scnns 28705

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-ne 2956  df-v 3452  df-dif 4047  df-sn 4585  df-nns 28707

This theorem is used by:  nnsgt0  28731  2ne0s  28812  expnnsval  28818  recut  28886  elreno2  28887  renegscl  28890  readdscl  28891  remulscllem1  28892  remulscl  28894