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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 4540 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 2147 | . . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1)) | |
2 | breq2 3941 | . . 3 ⊢ (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1)) | |
3 | 1, 2 | orbi12d 783 | . 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1))) |
4 | eqeq1 2147 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1)) | |
5 | breq2 3941 | . . 3 ⊢ (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦)) | |
6 | 4, 5 | orbi12d 783 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦))) |
7 | eqeq1 2147 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1)) | |
8 | breq2 3941 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1))) | |
9 | 7, 8 | orbi12d 783 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1)))) |
10 | eqeq1 2147 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1)) | |
11 | breq2 3941 | . . 3 ⊢ (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴)) | |
12 | 10, 11 | orbi12d 783 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴))) |
13 | eqid 2140 | . . 3 ⊢ 1 = 1 | |
14 | 13 | orci 721 | . 2 ⊢ (1 = 1 ∨ 1 < 1) |
15 | nngt0 8769 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦) | |
16 | nnre 8751 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
17 | 1re 7789 | . . . . . 6 ⊢ 1 ∈ ℝ | |
18 | ltaddpos2 8239 | . . . . . 6 ⊢ ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 𝑦 ↔ 1 < (𝑦 + 1))) | |
19 | 16, 17, 18 | sylancl 410 | . . . . 5 ⊢ (𝑦 ∈ ℕ → (0 < 𝑦 ↔ 1 < (𝑦 + 1))) |
20 | 15, 19 | mpbid 146 | . . . 4 ⊢ (𝑦 ∈ ℕ → 1 < (𝑦 + 1)) |
21 | 20 | olcd 724 | . . 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 8760 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 698 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 < clt 7824 ℕcn 8744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-inn 8745 |
This theorem is referenced by: nngt1ne1 8779 resqrexlemglsq 10826 |
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