Theorem nnne0s 28756

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 4753	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s )
2		df-nns 28734	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2882	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ≠ wne 2959   ∖ cdif 4048  {csn 4588   0_s c0s 28224  ℕ_0scn0s 28731  ℕ_scnns 28732

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-sn 4589  df-nns 28734

This theorem is used by:  nnsgt0  28758  2ne0s  28839  expnnsval  28845  recut  28913  elreno2  28914  renegscl  28917  readdscl  28918  remulscllem1  28919  remulscl  28921