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Theorem arch 8352
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.)
Assertion
Ref Expression
arch (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
Distinct variable group:   𝐴,𝑛

Proof of Theorem arch
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-arch 7157 . . 3 (𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
2 dfnn2 8108 . . . 4 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
32rexeqi 2555 . . 3 (∃𝑛 ∈ ℕ 𝐴 < 𝑛 ↔ ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
41, 3sylibr 132 . 2 (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
5 nnre 8113 . . . 4 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
6 ltxrlt 7245 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝐴 < 𝑛𝐴 < 𝑛))
75, 6sylan2 280 . . 3 ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐴 < 𝑛𝐴 < 𝑛))
87rexbidva 2366 . 2 (𝐴 ∈ ℝ → (∃𝑛 ∈ ℕ 𝐴 < 𝑛 ↔ ∃𝑛 ∈ ℕ 𝐴 < 𝑛))
94, 8mpbird 165 1 (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1434  {cab 2068  wral 2349  wrex 2350   cint 3644   class class class wbr 3793  (class class class)co 5543  cr 7042  1c1 7044   + caddc 7046   < cltrr 7047   < clt 7215  cn 8106
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1re 7132  ax-addrcl 7135  ax-arch 7157
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-xp 4377  df-pnf 7217  df-mnf 7218  df-ltxr 7220  df-inn 8107
This theorem is referenced by:  nnrecl  8353  bndndx  8354  btwnz  8547  expnbnd  9693  cvg1nlemres  10009  cvg1n  10010  resqrexlemga  10047  alzdvds  10399  dvdsbnd  10492
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