Theorem infpn2 16934

Description: There exist infinitely many prime numbers: the set of all primes 𝑆 is unbounded by infpn 16933, so by unben 16930 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 4062	. 2 ⊢ 𝑆 ⊆ ℕ
3		infpn 16933	. . . . 5 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
4		nnge1 12276	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 1 ≤ 𝑗)
5	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑗)
6		1re 11243	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
7		nnre 12255	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
8		nnre 12255	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
9		lelttr 11333	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑗 ∧ 𝑗 < 𝑘) → 1 < 𝑘))
10	6, 7, 8, 9	mp3an3an 1468	. . . . . . . . . 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 550	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑗 < 𝑘 → (𝑗 < 𝑘 ∧ 1 < 𝑘)))
13	12	anim1d 611	. . . . . . 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 468	. . . . . . 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	imbitrdi 251	. . . . . 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 3155	. . . . 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 5127	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (1 < 𝑛 ↔ 1 < 𝑘))
19		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 / 𝑚) = (𝑘 / 𝑚))
20	19	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 / 𝑚) ∈ ℕ ↔ (𝑘 / 𝑚) ∈ ℕ))
21		equequ2 2024	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑚 = 𝑛 ↔ 𝑚 = 𝑘))
22	21	orbi2d 915	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑚 = 1 ∨ 𝑚 = 𝑛) ↔ (𝑚 = 1 ∨ 𝑚 = 𝑘)))
23	20, 22	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)) ↔ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
24	23	ralbidv 3165	. . . . . . . . 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 632	. . . . . . . 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 3678	. . . . . . 7 ⊢ (𝑘 ∈ 𝑆 ↔ (𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
27	26	anbi1i 624	. . . . . 6 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑗 < 𝑘) ↔ ((𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))) ∧ 𝑗 < 𝑘))
28		anass 468	. . . . . 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 460	. . . . . . 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 623	. . . . . 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 297	. . . . 5 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑗 < 𝑘) ↔ (𝑘 ∈ ℕ ∧ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))))
32	31	rexbii2 3078	. . . 4 ⊢ (∃𝑘 ∈ 𝑆 𝑗 < 𝑘 ↔ ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
33	17, 32	sylibr 234	. . 3 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ 𝑆 𝑗 < 𝑘)
34	33	rgen 3052	. 2 ⊢ ∀𝑗 ∈ ℕ ∃𝑘 ∈ 𝑆 𝑗 < 𝑘
35		unben 16930	. 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 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  {crab 3419   ⊆ wss 3931   class class class wbr 5123  (class class class)co 7413   ≈ cen 8964  ℝcr 11136  1c1 11138   < clt 11277   ≤ cle 11278   / cdiv 11902  ℕcn 12248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-inf2 9663  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-div 11903  df-nn 12249  df-n0 12510  df-z 12597  df-uz 12861  df-seq 14025  df-fac 14296

This theorem is referenced by: (None)