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

Theorem infpn2 13057
Description: There exist infinitely many prime numbers: the set of all primes  S is unbounded by infpn 13056, so by unben 13053 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 3334 . . 3  |-  { n  e.  NN  |  ( 1  <  n  /\  A. m  e.  NN  (
( n  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  n ) ) ) }  C_  NN
31, 2eqsstri 3284 . 2  |-  S  C_  NN
4 infpn 13056 . . . . 5  |-  ( j  e.  NN  ->  E. k  e.  NN  ( j  < 
k  /\  A. m  e.  NN  ( ( k  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) )
5 nnge1 9862 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  1  <_  j )
65adantr 451 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  1  <_  j )
7 nnre 9843 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  j  e.  RR )
8 nnre 9843 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
9 1re 8927 . . . . . . . . . . . 12  |-  1  e.  RR
10 lelttr 9002 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  j  e.  RR  /\  k  e.  RR )  ->  (
( 1  <_  j  /\  j  <  k )  ->  1  <  k
) )
119, 10mp3an1 1264 . . . . . . . . . . 11  |-  ( ( j  e.  RR  /\  k  e.  RR )  ->  ( ( 1  <_ 
j  /\  j  <  k )  ->  1  <  k ) )
127, 8, 11syl2an 463 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( ( 1  <_ 
j  /\  j  <  k )  ->  1  <  k ) )
136, 12mpand 656 . . . . . . . . 9  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( j  <  k  ->  1  <  k ) )
1413ancld 536 . . . . . . . 8  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( j  <  k  ->  ( j  <  k  /\  1  <  k ) ) )
1514anim1d 547 . . . . . . 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 630 . . . . . . 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 217 . . . . . 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 2731 . . . . 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 14 . . . 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 4108 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  <  n  <->  1  <  k ) )
21 oveq1 5952 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
n  /  m )  =  ( k  /  m ) )
2221eleq1d 2424 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
( n  /  m
)  e.  NN  <->  ( k  /  m )  e.  NN ) )
23 equequ2 1686 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
m  =  n  <->  m  =  k ) )
2423orbi2d 682 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
( m  =  1  \/  m  =  n )  <->  ( m  =  1  \/  m  =  k ) ) )
2522, 24imbi12d 311 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( ( n  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  n ) )  <->  ( (
k  /  m )  e.  NN  ->  (
m  =  1  \/  m  =  k ) ) ) )
2625ralbidv 2639 . . . . . . . . 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 691 . . . . . . . 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 3001 . . . . . . 7  |-  ( k  e.  S  <->  ( k  e.  NN  /\  ( 1  <  k  /\  A. m  e.  NN  (
( k  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) ) )
2928anbi1i 676 . . . . . 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 630 . . . . . 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 437 . . . . . . 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 675 . . . . . 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 262 . . . . 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 2648 . . . 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 203 . . 3  |-  ( j  e.  NN  ->  E. k  e.  S  j  <  k )
3635rgen 2684 . 2  |-  A. j  e.  NN  E. k  e.  S  j  <  k
37 unben 13053 . 2  |-  ( ( S  C_  NN  /\  A. j  e.  NN  E. k  e.  S  j  <  k )  ->  S  ~~  NN )
383, 36, 37mp2an 653 1  |-  S  ~~  NN
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   {crab 2623    C_ wss 3228   class class class wbr 4104  (class class class)co 5945    ~~ cen 6948   RRcr 8826   1c1 8828    < clt 8957    <_ cle 8958    / cdiv 9513   NNcn 9836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-seq 11139  df-fac 11382
  Copyright terms: Public domain W3C validator