Theorem nnne0s 28508

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 4759	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s )
2		df-nns 28486	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2881	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

Syntax hints:   → wi 4   ∈ wcel 2143   ≠ wne 2958   ∖ cdif 3903  {csn 4590   0_s c0s 27976  ℕ_0scn0s 28483  ℕ_scnns 28484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3909  df-sn 4591  df-nns 28486

This theorem is referenced by:  nnsgt0  28510  2ne0s  28591  expnnsval  28597  recut  28665  elreno2  28666  renegscl  28669  readdscl  28670  remulscllem1  28671  remulscl  28673