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| Mirrors > Home > MPE Home > Th. List > nnne0s | Structured version Visualization version GIF version | ||
| Description: A surreal positive integer is non-zero. (Contributed by Scott Fenton, 15-Apr-2025.) | 
| Ref | Expression | 
|---|---|
| nnne0s | ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eldifsni 4789 | . 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s ) | |
| 2 | df-nns 28322 | . 2 ⊢ ℕs = (ℕ0s ∖ { 0s }) | |
| 3 | 1, 2 | eleq2s 2858 | 1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 {csn 4625 0s c0s 27868 ℕ0scnn0s 28319 ℕscnns 28320 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-v 3481 df-dif 3953 df-sn 4626 df-nns 28322 | 
| This theorem is referenced by: nnsgt0 28343 2ne0s 28405 expsnnval 28410 recut 28429 0reno 28430 renegscl 28431 readdscl 28432 remulscllem1 28433 remulscl 28435 | 
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