Theorem elnns 28124

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 28104	. . 3 ⊢ ℕs = (ℕ0s ∖ { 0s })
2	1	eleq2i 2817	. 2 ⊢ (𝐴 ∈ ℕs ↔ 𝐴 ∈ (ℕ0s ∖ { 0s }))
3		eldifsn 4782	. 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 2098   ≠ wne 2932   ∖ cdif 3937  {csn 4620   0_s c0s 27671  ℕ_0scnn0s 28101  ℕ_scnns 28102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3943  df-sn 4621  df-nns 28104

This theorem is referenced by:  elnns2  28125