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Theorem axarch 7868
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 8897 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 7908. (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 7805 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐴)
21biimpi 120 . 2 (𝐴 ∈ ℝ → ∃𝑧R𝑧, 0R⟩ = 𝐴)
3 archsr 7759 . . . 4 (𝑧R → ∃𝑤N 𝑧 <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
43ad2antrl 490 . . 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 536 . . . . 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 531 . . . . . 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 7816 . . . . . 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 134 . . . . 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 4024 . . . 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 7825 . . . . . 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 490 . . . . 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 110 . . . . . 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 4012 . . . . 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 2845 . . . 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 2599 . 2 ((𝐴 ∈ ℝ ∧ (𝑧R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
172, 16rexlimddv 2599 1 (𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  cop 3594   cint 3842   class class class wbr 4000  (class class class)co 5868  1oc1o 6403  [cec 6526  Ncnpi 7249   ~Q ceq 7256   <Q cltq 7262  1Pc1p 7269   +P cpp 7270   ~R cer 7273  Rcnr 7274  0Rc0r 7275   <R cltr 7280  cr 7788  1c1 7790   + caddc 7792   < cltrr 7793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4285  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-2o 6411  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-pli 7282  df-mi 7283  df-lti 7284  df-plpq 7321  df-mpq 7322  df-enq 7324  df-nqqs 7325  df-plqqs 7326  df-mqqs 7327  df-1nqqs 7328  df-rq 7329  df-ltnqqs 7330  df-enq0 7401  df-nq0 7402  df-0nq0 7403  df-plq0 7404  df-mq0 7405  df-inp 7443  df-i1p 7444  df-iplp 7445  df-iltp 7447  df-enr 7703  df-nr 7704  df-plr 7705  df-ltr 7707  df-0r 7708  df-1r 7709  df-c 7795  df-1 7797  df-r 7799  df-add 7800  df-lt 7802
This theorem is referenced by: (None)
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