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Theorem arch 8150
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 6984 . . 3 (𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
2 dfnn2 7892 . . . 4 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
32rexeqi 2507 . . 3 (∃𝑛 ∈ ℕ 𝐴 < 𝑛 ↔ ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
41, 3sylibr 137 . 2 (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
5 nnre 7897 . . . 4 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
6 ltxrlt 7065 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝐴 < 𝑛𝐴 < 𝑛))
75, 6sylan2 270 . . 3 ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐴 < 𝑛𝐴 < 𝑛))
87rexbidva 2320 . 2 (𝐴 ∈ ℝ → (∃𝑛 ∈ ℕ 𝐴 < 𝑛 ↔ ∃𝑛 ∈ ℕ 𝐴 < 𝑛))
94, 8mpbird 156 1 (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wcel 1393  {cab 2026  wral 2303  wrex 2304   cint 3612   class class class wbr 3761  (class class class)co 5499  cr 6869  1c1 6871   + caddc 6873   < cltrr 6874   < clt 7040  cn 7890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-cnex 6956  ax-resscn 6957  ax-1re 6959  ax-addrcl 6962  ax-arch 6984
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-br 3762  df-opab 3816  df-xp 4338  df-pnf 7042  df-mnf 7043  df-ltxr 7045  df-inn 7891
This theorem is referenced by:  nnrecl  8151  bndndx  8152  btwnz  8329  expnbnd  9250  cvg1nlemres  9462  cvg1n  9463  resqrexlemga  9499
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