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| Mirrors > Home > MPE Home > Th. List > eln0s | Structured version Visualization version GIF version | ||
| Description: A non-negative surreal integer is zero or a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.) |
| Ref | Expression |
|---|---|
| eln0s | ⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs ∨ 𝐴 = 0s )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.1 897 | . . . 4 ⊢ (¬ 𝐴 = 0s ∨ 𝐴 = 0s ) | |
| 2 | df-ne 2941 | . . . . 5 ⊢ (𝐴 ≠ 0s ↔ ¬ 𝐴 = 0s ) | |
| 3 | 2 | orbi1i 914 | . . . 4 ⊢ ((𝐴 ≠ 0s ∨ 𝐴 = 0s ) ↔ (¬ 𝐴 = 0s ∨ 𝐴 = 0s )) |
| 4 | 1, 3 | mpbir 231 | . . 3 ⊢ (𝐴 ≠ 0s ∨ 𝐴 = 0s ) |
| 5 | ordir 1009 | . . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ) ∧ (𝐴 ≠ 0s ∨ 𝐴 = 0s ))) | |
| 6 | 4, 5 | mpbiran2 710 | . 2 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s )) |
| 7 | elnns 28343 | . . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s )) | |
| 8 | 7 | orbi1i 914 | . 2 ⊢ ((𝐴 ∈ ℕs ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s )) |
| 9 | orc 868 | . . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s )) | |
| 10 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s) | |
| 11 | id 22 | . . . . 5 ⊢ (𝐴 = 0s → 𝐴 = 0s ) | |
| 12 | 0n0s 28334 | . . . . 5 ⊢ 0s ∈ ℕ0s | |
| 13 | 11, 12 | eqeltrdi 2849 | . . . 4 ⊢ (𝐴 = 0s → 𝐴 ∈ ℕ0s) |
| 14 | 10, 13 | jaoi 858 | . . 3 ⊢ ((𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ) → 𝐴 ∈ ℕ0s) |
| 15 | 9, 14 | impbii 209 | . 2 ⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s )) |
| 16 | 6, 8, 15 | 3bitr4ri 304 | 1 ⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs ∨ 𝐴 = 0s )) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-0s 27869 df-n0s 28320 df-nns 28321 |
| This theorem is referenced by: n0zs 28375 elzs2 28385 elznns 28388 expsp1 28412 |
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