Theorem axarch 7823

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 8850 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 7863. (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 7760	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐴)
2	1	biimpi 119	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐴)
3		archsr 7714	. . . 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 482	. . 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 526	. . . . 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 522	. . . . . 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 7771	. . . . . 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 133	. . . . 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 4000	. . . 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 7780	. . . . . 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 482	. . . . 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 109	. . . . . 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 3988	. . . . 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 2829	. . . 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 2586	. 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
17	2, 16	rexlimddv 2586	1 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1342   ∈ wcel 2135  {cab 2150  ∀wral 2442  ∃wrex 2443  ⟨cop 3573  ∩ cint 3818   class class class wbr 3976  (class class class)co 5836  1_oc1o 6368  [cec 6490  Ncnpi 7204   ~_Q ceq 7211   <_Q cltq 7217  1_Pc1p 7224   +_P cpp 7225   ~_R cer 7228  Rcnr 7229  0_Rc0r 7230   <_R cltr 7235  ℝcr 7743  1c1 7745   + caddc 7747   <_ℝ cltrr 7748

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-eprel 4261  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-1o 6375  df-2o 6376  df-oadd 6379  df-omul 6380  df-er 6492  df-ec 6494  df-qs 6498  df-ni 7236  df-pli 7237  df-mi 7238  df-lti 7239  df-plpq 7276  df-mpq 7277  df-enq 7279  df-nqqs 7280  df-plqqs 7281  df-mqqs 7282  df-1nqqs 7283  df-rq 7284  df-ltnqqs 7285  df-enq0 7356  df-nq0 7357  df-0nq0 7358  df-plq0 7359  df-mq0 7360  df-inp 7398  df-i1p 7399  df-iplp 7400  df-iltp 7402  df-enr 7658  df-nr 7659  df-plr 7660  df-ltr 7662  df-0r 7663  df-1r 7664  df-c 7750  df-1 7752  df-r 7754  df-add 7755  df-lt 7757

This theorem is referenced by: (None)