Theorem elnns 28269

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 28246	. . 3 ⊢ ℕs = (ℕ0s ∖ { 0s })
2	1	eleq2i 2825	. 2 ⊢ (𝐴 ∈ ℕs ↔ 𝐴 ∈ (ℕ0s ∖ { 0s }))
3		eldifsn 4737	. 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 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:  elnns2  28270  nnsge1  28272  eln0s  28288  n0subs2  28291  dfnns2  28298