Theorem elnns 28332

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 28307	. . 3 ⊢ ℕs = (ℕ0s ∖ { 0s })
2	1	eleq2i 2828	. 2 ⊢ (𝐴 ∈ ℕs ↔ 𝐴 ∈ (ℕ0s ∖ { 0s }))
3		eldifsn 4731	. 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 2114   ≠ wne 2932   ∖ cdif 3886  {csn 4567   0_s c0s 27797  ℕ_0scn0s 28304  ℕ_scnns 28305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3431  df-dif 3892  df-sn 4568  df-nns 28307

This theorem is referenced by:  elnns2  28333  nnsge1  28335  eln0s  28353  n0subs2  28356  dfnns2  28364