Theorem nnne0s 28266

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 4741	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s )
2		df-nns 28246	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2851	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

Syntax hints:   → wi 4   ∈ wcel 2113   ≠ wne 2929   ∖ cdif 3895  {csn 4575   0_s c0s 27767  ℕ_0scnn0s 28243  ℕ_scnns 28244

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 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-v 3439  df-dif 3901  df-sn 4576  df-nns 28246

This theorem is referenced by:  nnsgt0  28268  2ne0s  28344  expsnnval  28350  recut  28399  0reno  28400  renegscl  28401  readdscl  28402  remulscllem1  28403  remulscl  28405