ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  axarch GIF version

Theorem axarch 7022
Description: Archimedean axiom. The Archimedean property is more naturally stated once we have defined . Unless we find another way to state it, we'll just use the right hand side of dfnn2 7991 in stating what we mean by "natural number" in the context of this axiom.

This construction-dependent theorem should not be referenced directly; instead, use ax-arch 7060. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)

Assertion
Ref Expression
axarch (𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
Distinct variable group:   𝐴,𝑛,𝑥,𝑦

Proof of Theorem axarch
Dummy variables 𝑙 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 6962 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐴)
21biimpi 117 . 2 (𝐴 ∈ ℝ → ∃𝑧R𝑧, 0R⟩ = 𝐴)
3 archsr 6923 . . . 4 (𝑧R → ∃𝑤N 𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
43ad2antrl 467 . . 3 ((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑤N 𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
5 simplrr 496 . . . . 5 (((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨𝑧, 0R⟩ = 𝐴)
6 simprr 492 . . . . . 6 (((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → 𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
7 ltresr 6972 . . . . . 6 (⟨𝑧, 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ 𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
86, 7sylibr 141 . . . . 5 (((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨𝑧, 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
95, 8eqbrtrrd 3813 . . . 4 (((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → 𝐴 < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
10 pitonn 6981 . . . . . 6 (𝑤N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
1110ad2antrl 467 . . . . 5 (((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
12 simpr 107 . . . . . 6 ((((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ∧ 𝑛 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) → 𝑛 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
1312breq2d 3803 . . . . 5 ((((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ∧ 𝑛 = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) → (𝐴 < 𝑛𝐴 < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
1411, 13rspcedv 2677 . . . 4 (((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → (𝐴 < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛))
159, 14mpd 13 . . 3 (((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤N𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
164, 15rexlimddv 2454 . 2 ((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
172, 16rexlimddv 2454 1 (𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  cop 3405   cint 3642   class class class wbr 3791  (class class class)co 5539  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   ~Q ceq 6434   <Q cltq 6440  1Pc1p 6447   +P cpp 6448   ~R cer 6451  Rcnr 6452  0Rc0r 6453   <R cltr 6458  cr 6945  1c1 6947   + caddc 6949   < cltrr 6950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-iltp 6625  df-enr 6868  df-nr 6869  df-plr 6870  df-ltr 6872  df-0r 6873  df-1r 6874  df-c 6952  df-1 6954  df-r 6956  df-add 6957  df-lt 6959
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator