Theorem infpn2 16305

Description: There exist infinitely many prime numbers: the set of all primes 𝑆 is unbounded by infpn 16304, so by unben 16301 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.)

Hypothesis

Ref	Expression
infpn2.1	⊢ 𝑆 = {𝑛 ∈ ℕ ∣ (1 < 𝑛 ∧ ∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)))}

Assertion

Ref	Expression
infpn2	⊢ 𝑆 ≈ ℕ

Distinct variable group:   𝑚,𝑛

Allowed substitution hints:   𝑆(𝑚,𝑛)

Proof of Theorem infpn2

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infpn2.1	. . 3 ⊢ 𝑆 = {𝑛 ∈ ℕ ∣ (1 < 𝑛 ∧ ∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)))}
2	1	ssrab3 3987	. 2 ⊢ 𝑆 ⊆ ℕ
3		infpn 16304	. . . . 5 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
4		nnge1 11703	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 1 ≤ 𝑗)
5	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑗)
6		1re 10680	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
7		nnre 11682	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
8		nnre 11682	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
9		lelttr 10770	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑗 ∧ 𝑗 < 𝑘) → 1 < 𝑘))
10	6, 7, 8, 9	mp3an3an 1465	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → ((1 ≤ 𝑗 ∧ 𝑗 < 𝑘) → 1 < 𝑘))
11	5, 10	mpand 695	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑗 < 𝑘 → 1 < 𝑘))
12	11	ancld 555	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑗 < 𝑘 → (𝑗 < 𝑘 ∧ 1 < 𝑘)))
13	12	anim1d 614	. . . . . . 7 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → ((𝑗 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))) → ((𝑗 < 𝑘 ∧ 1 < 𝑘) ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
14		anass 473	. . . . . . 7 ⊢ (((𝑗 < 𝑘 ∧ 1 < 𝑘) ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))) ↔ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
15	13, 14	syl6ib 254	. . . . . 6 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → ((𝑗 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))) → (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))))
16	15	reximdva 3199	. . . . 5 ⊢ (𝑗 ∈ ℕ → (∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))) → ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))))
17	3, 16	mpd 15	. . . 4 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
18		breq2 5037	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (1 < 𝑛 ↔ 1 < 𝑘))
19		oveq1 7158	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 / 𝑚) = (𝑘 / 𝑚))
20	19	eleq1d 2837	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 / 𝑚) ∈ ℕ ↔ (𝑘 / 𝑚) ∈ ℕ))
21		equequ2 2034	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑚 = 𝑛 ↔ 𝑚 = 𝑘))
22	21	orbi2d 914	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑚 = 1 ∨ 𝑚 = 𝑛) ↔ (𝑚 = 1 ∨ 𝑚 = 𝑘)))
23	20, 22	imbi12d 349	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)) ↔ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
24	23	ralbidv 3127	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)) ↔ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
25	18, 24	anbi12d 634	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((1 < 𝑛 ∧ ∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛))) ↔ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
26	25, 1	elrab2 3606	. . . . . . 7 ⊢ (𝑘 ∈ 𝑆 ↔ (𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
27	26	anbi1i 627	. . . . . 6 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑗 < 𝑘) ↔ ((𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))) ∧ 𝑗 < 𝑘))
28		anass 473	. . . . . 6 ⊢ (((𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))) ∧ 𝑗 < 𝑘) ↔ (𝑘 ∈ ℕ ∧ ((1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))) ∧ 𝑗 < 𝑘)))
29		ancom 465	. . . . . . 7 ⊢ (((1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))) ∧ 𝑗 < 𝑘) ↔ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
30	29	anbi2i 626	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ ((1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))) ∧ 𝑗 < 𝑘)) ↔ (𝑘 ∈ ℕ ∧ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))))
31	27, 28, 30	3bitri 301	. . . . 5 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑗 < 𝑘) ↔ (𝑘 ∈ ℕ ∧ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))))
32	31	rexbii2 3174	. . . 4 ⊢ (∃𝑘 ∈ 𝑆 𝑗 < 𝑘 ↔ ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
33	17, 32	sylibr 237	. . 3 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ 𝑆 𝑗 < 𝑘)
34	33	rgen 3081	. 2 ⊢ ∀𝑗 ∈ ℕ ∃𝑘 ∈ 𝑆 𝑗 < 𝑘
35		unben 16301	. 2 ⊢ ((𝑆 ⊆ ℕ ∧ ∀𝑗 ∈ ℕ ∃𝑘 ∈ 𝑆 𝑗 < 𝑘) → 𝑆 ≈ ℕ)
36	2, 34, 35	mp2an 692	1 ⊢ 𝑆 ≈ ℕ

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  {crab 3075   ⊆ wss 3859   class class class wbr 5033  (class class class)co 7151   ≈ cen 8525  ℝcr 10575  1c1 10577   < clt 10714   ≤ cle 10715   / cdiv 11336  ℕcn 11675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-inf2 9138  ax-cnex 10632  ax-resscn 10633  ax-1cn 10634  ax-icn 10635  ax-addcl 10636  ax-addrcl 10637  ax-mulcl 10638  ax-mulrcl 10639  ax-mulcom 10640  ax-addass 10641  ax-mulass 10642  ax-distr 10643  ax-i2m1 10644  ax-1ne0 10645  ax-1rid 10646  ax-rnegex 10647  ax-rrecex 10648  ax-cnre 10649  ax-pre-lttri 10650  ax-pre-lttrn 10651  ax-pre-ltadd 10652  ax-pre-mulgt0 10653

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10716  df-mnf 10717  df-xr 10718  df-ltxr 10719  df-le 10720  df-sub 10911  df-neg 10912  df-div 11337  df-nn 11676  df-n0 11936  df-z 12022  df-uz 12284  df-seq 13420  df-fac 13685

This theorem is referenced by: (None)