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Theorem arch 8561
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 7366 . . 3 (𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
2 dfnn2 8317 . . . 4 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
32rexeqi 2560 . . 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 8322 . . . 4 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
6 ltxrlt 7454 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝐴 < 𝑛𝐴 < 𝑛))
75, 6sylan2 280 . . 3 ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐴 < 𝑛𝐴 < 𝑛))
87rexbidva 2371 . 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 2069  wral 2353  wrex 2354   cint 3662   class class class wbr 3811  (class class class)co 5590  cr 7251  1c1 7253   + caddc 7255   < cltrr 7256   < clt 7424  cn 8315
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 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-cnex 7338  ax-resscn 7339  ax-1re 7341  ax-addrcl 7344  ax-arch 7366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-xp 4406  df-pnf 7426  df-mnf 7427  df-ltxr 7429  df-inn 8316
This theorem is referenced by:  nnrecl  8562  bndndx  8563  btwnz  8760  expnbnd  9911  cvg1nlemres  10244  cvg1n  10245  resqrexlemga  10282  alzdvds  10634  dvdsbnd  10727
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