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Mirrors > Home > MPE Home > Th. List > elnns | Structured version Visualization version GIF version |
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 28336 | . . 3 ⊢ ℕs = (ℕ0s ∖ { 0s }) | |
2 | 1 | eleq2i 2831 | . 2 ⊢ (𝐴 ∈ ℕs ↔ 𝐴 ∈ (ℕ0s ∖ { 0s })) |
3 | eldifsn 4791 | . 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 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: elnns2 28359 nnsge1 28361 eln0s 28373 dfnns2 28377 |
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