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Mirrors > Home > ILE Home > Th. List > nn1gt1 | GIF version |
Description: A positive integer is either one or greater than one. This is for ℕ; 0elnn 4615 is a similar theorem for ω (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) |
Ref | Expression |
---|---|
nn1gt1 | ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2184 | . . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1)) | |
2 | breq2 4004 | . . 3 ⊢ (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1)) | |
3 | 1, 2 | orbi12d 793 | . 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1))) |
4 | eqeq1 2184 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1)) | |
5 | breq2 4004 | . . 3 ⊢ (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦)) | |
6 | 4, 5 | orbi12d 793 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦))) |
7 | eqeq1 2184 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1)) | |
8 | breq2 4004 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1))) | |
9 | 7, 8 | orbi12d 793 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1)))) |
10 | eqeq1 2184 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1)) | |
11 | breq2 4004 | . . 3 ⊢ (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴)) | |
12 | 10, 11 | orbi12d 793 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴))) |
13 | eqid 2177 | . . 3 ⊢ 1 = 1 | |
14 | 13 | orci 731 | . 2 ⊢ (1 = 1 ∨ 1 < 1) |
15 | nngt0 8930 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦) | |
16 | nnre 8912 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
17 | 1re 7944 | . . . . . 6 ⊢ 1 ∈ ℝ | |
18 | ltaddpos2 8397 | . . . . . 6 ⊢ ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 𝑦 ↔ 1 < (𝑦 + 1))) | |
19 | 16, 17, 18 | sylancl 413 | . . . . 5 ⊢ (𝑦 ∈ ℕ → (0 < 𝑦 ↔ 1 < (𝑦 + 1))) |
20 | 15, 19 | mpbid 147 | . . . 4 ⊢ (𝑦 ∈ ℕ → 1 < (𝑦 + 1)) |
21 | 20 | olcd 734 | . . 3 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))) |
22 | 21 | a1d 22 | . 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 = 1 ∨ 1 < 𝑦) → ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1)))) |
23 | 3, 6, 9, 12, 14, 22 | nnind 8921 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 (class class class)co 5869 ℝcr 7798 0cc0 7799 1c1 7800 + caddc 7802 < clt 7979 ℕcn 8905 |
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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-ltadd 7915 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-xp 4629 df-cnv 4631 df-iota 5174 df-fv 5220 df-ov 5872 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-inn 8906 |
This theorem is referenced by: nngt1ne1 8940 resqrexlemglsq 11012 |
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