Theorem nnne0s 28488

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 28466	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2883	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

Syntax hints:   → wi 4   ∈ wcel 2145   ≠ wne 2960   ∖ cdif 3904  {csn 4585   0_s c0s 27956  ℕ_0scn0s 28463  ℕ_scnns 28464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-sn 4586  df-nns 28466

This theorem is referenced by:  nnsgt0  28490  2ne0s  28571  expnnsval  28577  recut  28645  elreno2  28646  renegscl  28649  readdscl  28650  remulscllem1  28651  remulscl  28653