Theorem nnne0s 28354

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 4730	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s )
2		df-nns 28332	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2858	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

Syntax hints:   → wi 4   ∈ wcel 2119   ≠ wne 2935   ∖ cdif 3887  {csn 4562   0_s c0s 27822  ℕ_0scn0s 28329  ℕ_scnns 28330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-v 3434  df-dif 3893  df-sn 4563  df-nns 28332

This theorem is referenced by:  nnsgt0  28356  2ne0s  28437  expnnsval  28443  recut  28511  elreno2  28512  renegscl  28515  readdscl  28516  remulscllem1  28517  remulscl  28519