Theorem infpn2 17005

Description: There exist infinitely many prime numbers: the set of all primes 𝑆 is unbounded by infpn 17004, so by unben 17001 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 4030	. 2 ⊢ 𝑆 ⊆ ℕ
3		infpn 17004	. . . . 5 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
4		nnge1 12288	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 1 ≤ 𝑗)
5	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑗)
6		1re 11232	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
7		nnre 12264	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
8		nnre 12264	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
9		lelttr 11324	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑗 ∧ 𝑗 < 𝑘) → 1 < 𝑘))
10	6, 7, 8, 9	mp3an3an 1496	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → ((1 ≤ 𝑗 ∧ 𝑗 < 𝑘) → 1 < 𝑘))
11	5, 10	mpand 708	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑗 < 𝑘 → 1 < 𝑘))
12	11	ancld 560	. . . . . . . 8 ⊢ ((𝑗 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑗 < 𝑘 → (𝑗 < 𝑘 ∧ 1 < 𝑘)))
13	12	anim1d 623	. . . . . . 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 474	. . . . . . 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 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 3175	. . . . 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 16	. . . 4 ⊢ (𝑗 ∈ ℕ → ∃𝑘 ∈ ℕ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
18		breq2 5107	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (1 < 𝑛 ↔ 1 < 𝑘))
19		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 / 𝑚) = (𝑘 / 𝑚))
20	19	eleq1d 2845	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 / 𝑚) ∈ ℕ ↔ (𝑘 / 𝑚) ∈ ℕ))
21		equequ2 2059	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑚 = 𝑛 ↔ 𝑚 = 𝑘))
22	21	orbi2d 929	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑚 = 1 ∨ 𝑚 = 𝑛) ↔ (𝑚 = 1 ∨ 𝑚 = 𝑘)))
23	20, 22	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)) ↔ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))
24	23	ralbidv 3185	. . . . . . . . 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 644	. . . . . . . 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 3649	. . . . . . 7 ⊢ (𝑘 ∈ 𝑆 ↔ (𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))))
27	26	anbi1i 636	. . . . . 6 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑗 < 𝑘) ↔ ((𝑘 ∈ ℕ ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘)))) ∧ 𝑗 < 𝑘))
28		anass 474	. . . . . 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 466	. . . . . . 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 635	. . . . . 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 300	. . . . 5 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑗 < 𝑘) ↔ (𝑘 ∈ ℕ ∧ (𝑗 < 𝑘 ∧ (1 < 𝑘 ∧ ∀𝑚 ∈ ℕ ((𝑘 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑘))))))
32	31	rexbii2 3105	. . . 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 3078	. 2 ⊢ ∀𝑗 ∈ ℕ ∃𝑘 ∈ 𝑆 𝑗 < 𝑘
35		unben 17001	. 2 ⊢ ((𝑆 ⊆ ℕ ∧ ∀𝑗 ∈ ℕ ∃𝑘 ∈ 𝑆 𝑗 < 𝑘) → 𝑆 ≈ ℕ)
36	2, 34, 35	mp2an 705	1 ⊢ 𝑆 ≈ ℕ

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3899   class class class wbr 5103  (class class class)co 7413   ≈ cen 8949  ℝcr 11123  1c1 11125   < clt 11267   ≤ cle 11268   / cdiv 11895  ℕcn 12257

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-n0 12529  df-z 12616  df-uz 12888  df-seq 14066  df-fac 14338

This theorem is used by: (None)