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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 4795 | . 2 ⊢ (𝐴 ∈ (ℕ0s ∖ { 0s }) → 𝐴 ≠ 0s ) | |
2 | df-nns 28336 | . 2 ⊢ ℕs = (ℕ0s ∖ { 0s }) | |
3 | 1, 2 | eleq2s 2857 | 1 ⊢ (𝐴 ∈ ℕs → 𝐴 ≠ 0s ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 0s c0s 27882 ℕ0scnn0s 28333 ℕscnns 28334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-sn 4632 df-nns 28336 |
This theorem is referenced by: nnsgt0 28357 2ne0s 28419 expsnnval 28424 recut 28443 0reno 28444 renegscl 28445 readdscl 28446 remulscllem1 28447 remulscl 28449 |
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