Theorem nnne0s 28333

Description: A surreal positive integer is non-zero. (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 4746	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s )
2		df-nns 28311	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2854	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

Syntax hints:   → wi 4   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898  {csn 4580   0_s c0s 27801  ℕ_0scn0s 28308  ℕ_scnns 28309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3442  df-dif 3904  df-sn 4581  df-nns 28311

This theorem is referenced by:  nnsgt0  28335  2ne0s  28416  expnnsval  28422  recut  28490  elreno2  28491  renegscl  28494  readdscl  28495  remulscllem1  28496  remulscl  28498