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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 4532 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 2146 | . . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1)) | |
2 | breq2 3933 | . . 3 ⊢ (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1)) | |
3 | 1, 2 | orbi12d 782 | . 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1))) |
4 | eqeq1 2146 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1)) | |
5 | breq2 3933 | . . 3 ⊢ (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦)) | |
6 | 4, 5 | orbi12d 782 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦))) |
7 | eqeq1 2146 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1)) | |
8 | breq2 3933 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1))) | |
9 | 7, 8 | orbi12d 782 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1)))) |
10 | eqeq1 2146 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1)) | |
11 | breq2 3933 | . . 3 ⊢ (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴)) | |
12 | 10, 11 | orbi12d 782 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴))) |
13 | eqid 2139 | . . 3 ⊢ 1 = 1 | |
14 | 13 | orci 720 | . 2 ⊢ (1 = 1 ∨ 1 < 1) |
15 | nngt0 8745 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦) | |
16 | nnre 8727 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
17 | 1re 7765 | . . . . . 6 ⊢ 1 ∈ ℝ | |
18 | ltaddpos2 8215 | . . . . . 6 ⊢ ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 𝑦 ↔ 1 < (𝑦 + 1))) | |
19 | 16, 17, 18 | sylancl 409 | . . . . 5 ⊢ (𝑦 ∈ ℕ → (0 < 𝑦 ↔ 1 < (𝑦 + 1))) |
20 | 15, 19 | mpbid 146 | . . . 4 ⊢ (𝑦 ∈ ℕ → 1 < (𝑦 + 1)) |
21 | 20 | olcd 723 | . . 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 8736 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 < clt 7800 ℕcn 8720 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-iota 5088 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-inn 8721 |
This theorem is referenced by: nngt1ne1 8755 resqrexlemglsq 10794 |
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