Theorem axarch 8248

Description: Archimedean axiom. The Archimedean property is more naturally stated once we have defined ℕ. Unless we find another way to state it, we'll just use the right hand side of dfnn2 9285 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 8288. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
axarch	⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 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.

Step	Hyp	Ref	Expression
1		elreal 8185	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐴)
2	1	biimpi 120	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐴)
3		archsr 8139	. . . 4 ⊢ (𝑧 ∈ R → ∃𝑤 ∈ N 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
4	3	ad2antrl 494	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑤 ∈ N 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
5		simplrr 542	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨𝑧, 0R⟩ = 𝐴)
6		simprr 537	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
7		ltresr 8196	. . . . . 6 ⊢ (⟨𝑧, 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
8	6, 7	sylibr 134	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨𝑧, 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
9	5, 8	eqbrtrrd 4149	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → 𝐴 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
10		pitonn 8205	. . . . . 6 ⊢ (𝑤 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
11	10	ad2antrl 494	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
12		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ∧ 𝑛 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) → 𝑛 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
13	12	breq2d 4137	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ∧ 𝑛 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) → (𝐴 <ℝ 𝑛 ↔ 𝐴 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
14	11, 13	rspcedv 2933	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → (𝐴 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛))
15	9, 14	mpd 13	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
16	4, 15	rexlimddv 2673	. 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
17	2, 16	rexlimddv 2673	1 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  ⟨cop 3708  ∩ cint 3965   class class class wbr 4125  (class class class)co 6075  1_oc1o 6670  [cec 6795  Ncnpi 7629   ~_Q ceq 7636   <_Q cltq 7642  1_Pc1p 7649   +_P cpp 7650   ~_R cer 7653  Rcnr 7654  0_Rc0r 7655   <_R cltr 7660  ℝcr 8168  1c1 8170   + caddc 8172   <_ℝ cltrr 8173

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-i1p 7824  df-iplp 7825  df-iltp 7827  df-enr 8083  df-nr 8084  df-plr 8085  df-ltr 8087  df-0r 8088  df-1r 8089  df-c 8175  df-1 8177  df-r 8179  df-add 8180  df-lt 8182

This theorem is referenced by: (None)