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Theorem infpn2 14307
Description: There exist infinitely many prime numbers: the set of all primes  S is unbounded by infpn 14306, so by unben 14303 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.
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 3590 . . 3  |-  { n  e.  NN  |  ( 1  <  n  /\  A. m  e.  NN  (
( n  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  n ) ) ) }  C_  NN
31, 2eqsstri 3539 . 2  |-  S  C_  NN
4 infpn 14306 . . . . 5  |-  ( j  e.  NN  ->  E. k  e.  NN  ( j  < 
k  /\  A. m  e.  NN  ( ( k  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) )
5 nnge1 10574 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  1  <_  j )
65adantr 465 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  1  <_  j )
7 nnre 10555 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  j  e.  RR )
8 nnre 10555 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
9 1re 9607 . . . . . . . . . . . 12  |-  1  e.  RR
10 lelttr 9687 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  j  e.  RR  /\  k  e.  RR )  ->  (
( 1  <_  j  /\  j  <  k )  ->  1  <  k
) )
119, 10mp3an1 1311 . . . . . . . . . . 11  |-  ( ( j  e.  RR  /\  k  e.  RR )  ->  ( ( 1  <_ 
j  /\  j  <  k )  ->  1  <  k ) )
127, 8, 11syl2an 477 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( ( 1  <_ 
j  /\  j  <  k )  ->  1  <  k ) )
136, 12mpand 675 . . . . . . . . 9  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( j  <  k  ->  1  <  k ) )
1413ancld 553 . . . . . . . 8  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( j  <  k  ->  ( j  <  k  /\  1  <  k ) ) )
1514anim1d 564 . . . . . . 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 649 . . . . . . 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 2942 . . . . 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 4457 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  <  n  <->  1  <  k ) )
21 oveq1 6302 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
n  /  m )  =  ( k  /  m ) )
2221eleq1d 2536 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
( n  /  m
)  e.  NN  <->  ( k  /  m )  e.  NN ) )
23 equequ2 1748 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
m  =  n  <->  m  =  k ) )
2423orbi2d 701 . . . . . . . . . . 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 2906 . . . . . . . . 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 710 . . . . . . . 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 3268 . . . . . . 7  |-  ( k  e.  S  <->  ( k  e.  NN  /\  ( 1  <  k  /\  A. m  e.  NN  (
( k  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) ) )
2928anbi1i 695 . . . . . 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 649 . . . . . 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 450 . . . . . . 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 694 . . . . . 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 2967 . . . 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 2827 . 2  |-  A. j  e.  NN  E. k  e.  S  j  <  k
37 unben 14303 . 2  |-  ( ( S  C_  NN  /\  A. j  e.  NN  E. k  e.  S  j  <  k )  ->  S  ~~  NN )
383, 36, 37mp2an 672 1  |-  S  ~~  NN
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821    C_ wss 3481   class class class wbr 4453  (class class class)co 6295    ~~ cen 7525   RRcr 9503   1c1 9505    < clt 9640    <_ cle 9641    / cdiv 10218   NNcn 10548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-seq 12088  df-fac 12334
This theorem is referenced by: (None)
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