MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infpn2 Structured version   Unicode version

Theorem infpn2 13956
Description: There exist infinitely many prime numbers: the set of all primes  S is unbounded by infpn 13955, so by unben 13952 it is infinite. (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.
StepHypRef 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 3425 . . 3  |-  { n  e.  NN  |  ( 1  <  n  /\  A. m  e.  NN  (
( n  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  n ) ) ) }  C_  NN
31, 2eqsstri 3374 . 2  |-  S  C_  NN
4 infpn 13955 . . . . 5  |-  ( j  e.  NN  ->  E. k  e.  NN  ( j  < 
k  /\  A. m  e.  NN  ( ( k  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) )
5 nnge1 10335 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  1  <_  j )
65adantr 462 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  1  <_  j )
7 nnre 10316 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  j  e.  RR )
8 nnre 10316 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
9 1re 9372 . . . . . . . . . . . 12  |-  1  e.  RR
10 lelttr 9452 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  j  e.  RR  /\  k  e.  RR )  ->  (
( 1  <_  j  /\  j  <  k )  ->  1  <  k
) )
119, 10mp3an1 1294 . . . . . . . . . . 11  |-  ( ( j  e.  RR  /\  k  e.  RR )  ->  ( ( 1  <_ 
j  /\  j  <  k )  ->  1  <  k ) )
127, 8, 11syl2an 474 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( ( 1  <_ 
j  /\  j  <  k )  ->  1  <  k ) )
136, 12mpand 668 . . . . . . . . 9  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( j  <  k  ->  1  <  k ) )
1413ancld 548 . . . . . . . 8  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( j  <  k  ->  ( j  <  k  /\  1  <  k ) ) )
1514anim1d 559 . . . . . . 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 642 . . . . . . 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 ) ) ) ) )
1715, 16syl6ib 226 . . . . . 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 ) ) ) ) ) )
1817reximdva 2818 . . . . 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 ) ) ) ) ) )
194, 18mpd 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 4284 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  <  n  <->  1  <  k ) )
21 oveq1 6087 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
n  /  m )  =  ( k  /  m ) )
2221eleq1d 2499 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
( n  /  m
)  e.  NN  <->  ( k  /  m )  e.  NN ) )
23 equequ2 1736 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
m  =  n  <->  m  =  k ) )
2423orbi2d 694 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
( m  =  1  \/  m  =  n )  <->  ( m  =  1  \/  m  =  k ) ) )
2522, 24imbi12d 320 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( ( n  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  n ) )  <->  ( (
k  /  m )  e.  NN  ->  (
m  =  1  \/  m  =  k ) ) ) )
2625ralbidv 2725 . . . . . . . . 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 ) ) ) )
2720, 26anbi12d 703 . . . . . . . 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 ) ) ) ) )
2827, 1elrab2 3108 . . . . . . 7  |-  ( k  e.  S  <->  ( k  e.  NN  /\  ( 1  <  k  /\  A. m  e.  NN  (
( k  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) ) )
2928anbi1i 688 . . . . . 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 642 . . . . . 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 448 . . . . . . 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 ) ) ) ) )
3231anbi2i 687 . . . . . 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 ) ) ) ) ) )
3329, 30, 323bitri 271 . . . . 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 ) ) ) ) ) )
3433rexbii2 2734 . . . 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 ) ) ) ) )
3519, 34sylibr 212 . . 3  |-  ( j  e.  NN  ->  E. k  e.  S  j  <  k )
3635rgen 2771 . 2  |-  A. j  e.  NN  E. k  e.  S  j  <  k
37 unben 13952 . 2  |-  ( ( S  C_  NN  /\  A. j  e.  NN  E. k  e.  S  j  <  k )  ->  S  ~~  NN )
383, 36, 37mp2an 665 1  |-  S  ~~  NN
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706   {crab 2709    C_ wss 3316   class class class wbr 4280  (class class class)co 6080    ~~ cen 7295   RRcr 9268   1c1 9270    < clt 9405    <_ cle 9406    / cdiv 9980   NNcn 10309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-n0 10567  df-z 10634  df-uz 10849  df-seq 11790  df-fac 12035
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator