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Theorem arch 7954
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 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 6802 . . 3 (A ℝ → 𝑛 {x ∣ (1 x y x (y + 1) x)}A < 𝑛)
2 dfnn2 7697 . . . 4 ℕ = {x ∣ (1 x y x (y + 1) x)}
32rexeqi 2504 . . 3 (𝑛 A < 𝑛𝑛 {x ∣ (1 x y x (y + 1) x)}A < 𝑛)
41, 3sylibr 137 . 2 (A ℝ → 𝑛 A < 𝑛)
5 nnre 7702 . . . 4 (𝑛 ℕ → 𝑛 ℝ)
6 ltxrlt 6882 . . . 4 ((A 𝑛 ℝ) → (A < 𝑛A < 𝑛))
75, 6sylan2 270 . . 3 ((A 𝑛 ℕ) → (A < 𝑛A < 𝑛))
87rexbidva 2317 . 2 (A ℝ → (𝑛 A < 𝑛𝑛 A < 𝑛))
94, 8mpbird 156 1 (A ℝ → 𝑛 A < 𝑛)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  {cab 2023  wral 2300  wrex 2301   cint 3606   class class class wbr 3755  (class class class)co 5455  cr 6710  1c1 6712   + caddc 6714   < cltrr 6715   < clt 6857  cn 7695
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-1re 6777  ax-addrcl 6780  ax-arch 6802
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6859  df-mnf 6860  df-ltxr 6862  df-inn 7696
This theorem is referenced by:  nnrecl  7955  bndndx  7956  btwnz  8133  expnbnd  9025
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