Theorem infpn2 16082

Description: There exist infinitely many prime numbers: the set of all primes 𝑆 is unbounded by infpn 16081, so by unben 16078 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 3984	. 2 ⊢ 𝑆 ⊆ ℕ
3		infpn 16081	. . . . 5 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
4		nnge1 11519	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 1 ≤ 𝑗)
5	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑗)
6		1re 10494	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
7		nnre 11499	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
8		nnre 11499	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
9		lelttr 10584	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑗 ∧ 𝑗 < 𝑘) → 1 < 𝑘))
10	6, 7, 8, 9	mp3an3an 1459	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → ((1 ≤ 𝑗 ∧ 𝑗 < 𝑘) → 1 < 𝑘))
11	5, 10	mpand 691	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑗 < 𝑘 → 1 < 𝑘))
12	11	ancld 551	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑗 < 𝑘 → (𝑗 < 𝑘 ∧ 1 < 𝑘)))
13	12	anim1d 610	. . . . . . 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 469	. . . . . . 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 252	. . . . . 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 3239	. . . . 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 4972	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (1 < 𝑛 ↔ 1 < 𝑘))
19		oveq1 7030	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 / 𝑚) = (𝑘 / 𝑚))
20	19	eleq1d 2869	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 / 𝑚) ∈ ℕ ↔ (𝑘 / 𝑚) ∈ ℕ))
21		equequ2 2014	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑚 = 𝑛 ↔ 𝑚 = 𝑘))
22	21	orbi2d 910	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑚 = 1 ∨ 𝑚 = 𝑛) ↔ (𝑚 = 1 ∨ 𝑚 = 𝑘)))
23	20, 22	imbi12d 346	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)) ↔ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
24	23	ralbidv 3166	. . . . . . . . 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 630	. . . . . . . 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 3624	. . . . . . 7 ⊢ (𝑘 ∈ 𝑆 ↔ (𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
27	26	anbi1i 623	. . . . . 6 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑗 < 𝑘) ↔ ((𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))) ∧ 𝑗 < 𝑘))
28		anass 469	. . . . . 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 461	. . . . . . 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 622	. . . . . 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 298	. . . . 5 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑗 < 𝑘) ↔ (𝑘 ∈ ℕ ∧ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))))
32	31	rexbii2 3211	. . . 4 ⊢ (∃𝑘 ∈ 𝑆 𝑗 < 𝑘 ↔ ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
33	17, 32	sylibr 235	. . 3 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ 𝑆 𝑗 < 𝑘)
34	33	rgen 3117	. 2 ⊢ ∀𝑗 ∈ ℕ ∃𝑘 ∈ 𝑆 𝑗 < 𝑘
35		unben 16078	. 2 ⊢ ((𝑆 ⊆ ℕ ∧ ∀𝑗 ∈ ℕ ∃𝑘 ∈ 𝑆 𝑗 < 𝑘) → 𝑆 ≈ ℕ)
36	2, 34, 35	mp2an 688	1 ⊢ 𝑆 ≈ ℕ

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 842   = wceq 1525   ∈ wcel 2083  ∀wral 3107  ∃wrex 3108  {crab 3111   ⊆ wss 3865   class class class wbr 4968  (class class class)co 7023   ≈ cen 8361  ℝcr 10389  1c1 10391   < clt 10528   ≤ cle 10529   / cdiv 11151  ℕcn 11492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-n0 11752  df-z 11836  df-uz 12098  df-seq 13224  df-fac 13488

This theorem is referenced by: (None)