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| Description: Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025.) | 
| Ref | Expression | 
|---|---|
| elnns | ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s )) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-nns 28321 | . . 3 ⊢ ℕs = (ℕ0s ∖ { 0s }) | |
| 2 | 1 | eleq2i 2833 | . 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 )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 0s c0s 27867 ℕ0scnn0s 28318 ℕscnns 28319 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-sn 4627 df-nns 28321 | 
| This theorem is referenced by: elnns2 28344 nnsge1 28346 eln0s 28358 dfnns2 28362 | 
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