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Theorem axarch 7719
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 8742 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 7759. (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 7656 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐴)
21biimpi 119 . 2 (𝐴 ∈ ℝ → ∃𝑧R𝑧, 0R⟩ = 𝐴)
3 archsr 7610 . . . 4 (𝑧R → ∃𝑤N 𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
43ad2antrl 482 . . 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 526 . . . . 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 522 . . . . . 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 7667 . . . . . 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 133 . . . . 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 3956 . . . 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 7676 . . . . . 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 482 . . . . 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 109 . . . . . 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 3945 . . . . 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 2794 . . . 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 2555 . 2 ((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
172, 16rexlimddv 2555 1 (𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  {cab 2126  wral 2417  wrex 2418  cop 3531   cint 3775   class class class wbr 3933  (class class class)co 5778  1oc1o 6310  [cec 6431  Ncnpi 7100   ~Q ceq 7107   <Q cltq 7113  1Pc1p 7120   +P cpp 7121   ~R cer 7124  Rcnr 7125  0Rc0r 7126   <R cltr 7131  cr 7639  1c1 7641   + caddc 7643   < cltrr 7644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-2o 6318  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-mpq 7173  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-mqqs 7178  df-1nqqs 7179  df-rq 7180  df-ltnqqs 7181  df-enq0 7252  df-nq0 7253  df-0nq0 7254  df-plq0 7255  df-mq0 7256  df-inp 7294  df-i1p 7295  df-iplp 7296  df-iltp 7298  df-enr 7554  df-nr 7555  df-plr 7556  df-ltr 7558  df-0r 7559  df-1r 7560  df-c 7646  df-1 7648  df-r 7650  df-add 7651  df-lt 7653
This theorem is referenced by: (None)
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