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Mirrors > Home > ILE Home > Th. List > elnnnn0b | GIF version |
Description: The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
Ref | Expression |
---|---|
elnnnn0b | ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 9179 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | nngt0 8940 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
3 | 1, 2 | jca 306 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
4 | elnn0 9174 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
5 | ax-1 6 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0 < 𝑁 → 𝑁 ∈ ℕ)) | |
6 | breq2 4006 | . . . . . 6 ⊢ (𝑁 = 0 → (0 < 𝑁 ↔ 0 < 0)) | |
7 | 0re 7954 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 7 | ltnri 8046 | . . . . . . 7 ⊢ ¬ 0 < 0 |
9 | 8 | pm2.21i 646 | . . . . . 6 ⊢ (0 < 0 → 𝑁 ∈ ℕ) |
10 | 6, 9 | syl6bi 163 | . . . . 5 ⊢ (𝑁 = 0 → (0 < 𝑁 → 𝑁 ∈ ℕ)) |
11 | 5, 10 | jaoi 716 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (0 < 𝑁 → 𝑁 ∈ ℕ)) |
12 | 4, 11 | sylbi 121 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0 < 𝑁 → 𝑁 ∈ ℕ)) |
13 | 12 | imp 124 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 0 < 𝑁) → 𝑁 ∈ ℕ) |
14 | 3, 13 | impbii 126 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 class class class wbr 4002 0cc0 7808 < clt 7988 ℕcn 8915 ℕ0cn0 9172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-xp 4631 df-cnv 4633 df-iota 5177 df-fv 5223 df-ov 5875 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-inn 8916 df-n0 9173 |
This theorem is referenced by: elnnnn0c 9217 bccl2 10741 bezoutlemmain 11991 |
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