Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > arch | GIF version |
Description: Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.) |
Ref | Expression |
---|---|
arch | ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-arch 7863 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛) | |
2 | dfnn2 8850 | . . . 4 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
3 | 2 | rexeqi 2664 | . . 3 ⊢ (∃𝑛 ∈ ℕ 𝐴 <ℝ 𝑛 ↔ ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛) |
4 | 1, 3 | sylibr 133 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 <ℝ 𝑛) |
5 | nnre 8855 | . . . 4 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ) | |
6 | ltxrlt 7955 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝐴 < 𝑛 ↔ 𝐴 <ℝ 𝑛)) | |
7 | 5, 6 | sylan2 284 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐴 < 𝑛 ↔ 𝐴 <ℝ 𝑛)) |
8 | 7 | rexbidva 2461 | . 2 ⊢ (𝐴 ∈ ℝ → (∃𝑛 ∈ ℕ 𝐴 < 𝑛 ↔ ∃𝑛 ∈ ℕ 𝐴 <ℝ 𝑛)) |
9 | 4, 8 | mpbird 166 | 1 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2135 {cab 2150 ∀wral 2442 ∃wrex 2443 ∩ cint 3818 class class class wbr 3976 (class class class)co 5836 ℝcr 7743 1c1 7745 + caddc 7747 <ℝ cltrr 7748 < clt 7924 ℕcn 8848 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-xp 4604 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-inn 8849 |
This theorem is referenced by: nnrecl 9103 bndndx 9104 btwnz 9301 expnbnd 10567 cvg1nlemres 10913 cvg1n 10914 resqrexlemga 10951 fsum3cvg3 11323 divcnv 11424 efcllem 11586 alzdvds 11777 dvdsbnd 11874 |
Copyright terms: Public domain | W3C validator |