Theorem elnns 28255

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 28232	. . 3 ⊢ ℕs = (ℕ0s ∖ { 0s })
2	1	eleq2i 2820	. 2 ⊢ (𝐴 ∈ ℕs ↔ 𝐴 ∈ (ℕ0s ∖ { 0s }))
3		eldifsn 4740	. 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
4	2, 3	bitri 275	1 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3902  {csn 4579   0_s c0s 27754  ℕ_0scnn0s 28229  ℕ_scnns 28230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3440  df-dif 3908  df-sn 4580  df-nns 28232

This theorem is referenced by:  elnns2  28256  nnsge1  28258  eln0s  28274  n0subs2  28277  dfnns2  28284