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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 7095 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛) | |
2 | dfnn2 8041 | . . . 4 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} | |
3 | 2 | rexeqi 2554 | . . 3 ⊢ (∃𝑛 ∈ ℕ 𝐴 <ℝ 𝑛 ↔ ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛) |
4 | 1, 3 | sylibr 132 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 <ℝ 𝑛) |
5 | nnre 8046 | . . . 4 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ) | |
6 | ltxrlt 7178 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝐴 < 𝑛 ↔ 𝐴 <ℝ 𝑛)) | |
7 | 5, 6 | sylan2 280 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐴 < 𝑛 ↔ 𝐴 <ℝ 𝑛)) |
8 | 7 | rexbidva 2365 | . 2 ⊢ (𝐴 ∈ ℝ → (∃𝑛 ∈ ℕ 𝐴 < 𝑛 ↔ ∃𝑛 ∈ ℕ 𝐴 <ℝ 𝑛)) |
9 | 4, 8 | mpbird 165 | 1 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 {cab 2067 ∀wral 2348 ∃wrex 2349 ∩ cint 3636 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 1c1 6982 + caddc 6984 <ℝ cltrr 6985 < clt 7153 ℕcn 8039 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1re 7070 ax-addrcl 7073 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-inn 8040 |
This theorem is referenced by: nnrecl 8286 bndndx 8287 btwnz 8466 expnbnd 9596 cvg1nlemres 9871 cvg1n 9872 resqrexlemga 9909 alzdvds 10254 dvdsbnd 10348 |
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