Theorem infpn2 15617

Description: There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 15616, so by unben 15613 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.)

Hypothesis

Ref	Expression
infpn2.1	|- S = { n e. NN | ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) }

Assertion

Ref	Expression
infpn2	|- S ~~ NN

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 e. NN | ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) }
2		ssrab2 3687	. . 3 |- { n e. NN | ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) } C_ NN
3	1, 2	eqsstri 3635	. 2 |- S C_ NN
4		infpn 15616	. . . . 5 |- ( j e. NN -> E. k e. NN ( j < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) )
5		nnge1 11046	. . . . . . . . . . 11 |- ( j e. NN -> 1 <_ j )
6	5	adantr 481	. . . . . . . . . 10 |- ( ( j e. NN /\ k e. NN ) -> 1 <_ j )
7		nnre 11027	. . . . . . . . . . 11 |- ( j e. NN -> j e. RR )
8		nnre 11027	. . . . . . . . . . 11 |- ( k e. NN -> k e. RR )
9		1re 10039	. . . . . . . . . . . 12 |- 1 e. RR
10		lelttr 10128	. . . . . . . . . . . 12 |- ( ( 1 e. RR /\ j e. RR /\ k e. RR ) -> ( ( 1 <_ j /\ j < k ) -> 1 < k ) )
11	9, 10	mp3an1 1411	. . . . . . . . . . 11 |- ( ( j e. RR /\ k e. RR ) -> ( ( 1 <_ j /\ j < k ) -> 1 < k ) )
12	7, 8, 11	syl2an 494	. . . . . . . . . 10 |- ( ( j e. NN /\ k e. NN ) -> ( ( 1 <_ j /\ j < k ) -> 1 < k ) )
13	6, 12	mpand 711	. . . . . . . . 9 |- ( ( j e. NN /\ k e. NN ) -> ( j < k -> 1 < k ) )
14	13	ancld 576	. . . . . . . 8 |- ( ( j e. NN /\ k e. NN ) -> ( j < k -> ( j < k /\ 1 < k ) ) )
15	14	anim1d 588	. . . . . . 7 |- ( ( j e. NN /\ k e. NN ) -> ( ( j < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) -> ( ( j < k /\ 1 < k ) /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) )
16		anass 681	. . . . . . 7 |- ( ( ( j < k /\ 1 < k ) /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) <-> ( j < k /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) )
17	15, 16	syl6ib 241	. . . . . 6 |- ( ( j e. NN /\ k e. NN ) -> ( ( j < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) -> ( j < k /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) ) )
18	17	reximdva 3017	. . . . 5 |- ( j e. NN -> ( E. k e. NN ( j < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) -> E. k e. NN ( j < k /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) ) )
19	4, 18	mpd 15	. . . 4 |- ( j e. NN -> E. k e. NN ( j < k /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) )
20		breq2 4657	. . . . . . . . 9 |- ( n = k -> ( 1 < n <-> 1 < k ) )
21		oveq1 6657	. . . . . . . . . . . 12 |- ( n = k -> ( n / m ) = ( k / m ) )
22	21	eleq1d 2686	. . . . . . . . . . 11 |- ( n = k -> ( ( n / m ) e. NN <-> ( k / m ) e. NN ) )
23		equequ2 1953	. . . . . . . . . . . 12 |- ( n = k -> ( m = n <-> m = k ) )
24	23	orbi2d 738	. . . . . . . . . . 11 |- ( n = k -> ( ( m = 1 \/ m = n ) <-> ( m = 1 \/ m = k ) ) )
25	22, 24	imbi12d 334	. . . . . . . . . 10 |- ( n = k -> ( ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) <-> ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) )
26	25	ralbidv 2986	. . . . . . . . 9 |- ( n = k -> ( A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) <-> A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) )
27	20, 26	anbi12d 747	. . . . . . . 8 |- ( n = k -> ( ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) <-> ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) )
28	27, 1	elrab2 3366	. . . . . . 7 |- ( k e. S <-> ( k e. NN /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) )
29	28	anbi1i 731	. . . . . 6 |- ( ( k e. S /\ j < k ) <-> ( ( k e. NN /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) /\ j < k ) )
30		anass 681	. . . . . 6 |- ( ( ( k e. NN /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) /\ j < k ) <-> ( k e. NN /\ ( ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) /\ j < k ) ) )
31		ancom 466	. . . . . . 7 |- ( ( ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) /\ j < k ) <-> ( j < k /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) )
32	31	anbi2i 730	. . . . . 6 |- ( ( k e. NN /\ ( ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) /\ j < k ) ) <-> ( k e. NN /\ ( j < k /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) ) )
33	29, 30, 32	3bitri 286	. . . . 5 |- ( ( k e. S /\ j < k ) <-> ( k e. NN /\ ( j < k /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) ) )
34	33	rexbii2 3039	. . . 4 |- ( E. k e. S j < k <-> E. k e. NN ( j < k /\ ( 1 < k /\ A. m e. NN ( ( k / m ) e. NN -> ( m = 1 \/ m = k ) ) ) ) )
35	19, 34	sylibr 224	. . 3 |- ( j e. NN -> E. k e. S j < k )
36	35	rgen 2922	. 2 |- A. j e. NN E. k e. S j < k
37		unben 15613	. 2 |- ( ( S C_ NN /\ A. j e. NN E. k e. S j < k ) -> S ~~ NN )
38	3, 36, 37	mp2an 708	1 |- S ~~ NN

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   ~~ cen 7952   RRcr 9935   1c1 9937   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-fac 13061

This theorem is referenced by: (None)