Theorem elnns 28195

Description: Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
elnns	⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))

Proof of Theorem elnns

Step	Hyp	Ref	Expression
1		df-nns 28175	. . 3 ⊢ ℕs = (ℕ0s ∖ { 0s })
2	1	eleq2i 2820	. 2 ⊢ (𝐴 ∈ ℕs ↔ 𝐴 ∈ (ℕ0s ∖ { 0s }))
3		eldifsn 4786	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
4	2, 3	bitri 275	1 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2099   ≠ wne 2935   ∖ cdif 3941  {csn 4624   0_s c0s 27742  ℕ_0scnn0s 28172  ℕ_scnns 28173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-v 3471  df-dif 3947  df-sn 4625  df-nns 28175

This theorem is referenced by:  elnns2  28196