Theorem nnne0s 28610

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 4756	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s )
2		df-nns 28588	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
3	1, 2	eleq2s 2880	1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2957   ∖ cdif 3899  {csn 4587   0_s c0s 28078  ℕ_0scn0s 28585  ℕ_scnns 28586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-dif 3905  df-sn 4588  df-nns 28588

This theorem is used by:  nnsgt0  28612  2ne0s  28693  expnnsval  28699  recut  28767  elreno2  28768  renegscl  28771  readdscl  28772  remulscllem1  28773  remulscl  28775