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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 7664 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛) | |
2 | dfnn2 8632 | . . . 4 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
3 | 2 | rexeqi 2605 | . . 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 8637 | . . . 4 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ) | |
6 | ltxrlt 7754 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝐴 < 𝑛 ↔ 𝐴 <ℝ 𝑛)) | |
7 | 5, 6 | sylan2 282 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐴 < 𝑛 ↔ 𝐴 <ℝ 𝑛)) |
8 | 7 | rexbidva 2408 | . 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 1463 {cab 2101 ∀wral 2390 ∃wrex 2391 ∩ cint 3737 class class class wbr 3895 (class class class)co 5728 ℝcr 7546 1c1 7548 + caddc 7550 <ℝ cltrr 7551 < clt 7724 ℕcn 8630 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1re 7639 ax-addrcl 7642 ax-arch 7664 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-xp 4505 df-pnf 7726 df-mnf 7727 df-ltxr 7729 df-inn 8631 |
This theorem is referenced by: nnrecl 8879 bndndx 8880 btwnz 9074 expnbnd 10308 cvg1nlemres 10649 cvg1n 10650 resqrexlemga 10687 fsum3cvg3 11057 divcnv 11158 efcllem 11216 alzdvds 11400 dvdsbnd 11493 |
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