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Theorem infpn2 12960
Description: There exist infinitely many prime numbers: the set of all primes  S is unbounded by infpn 12959, so by unben 12956 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 3258 . . 3  |-  { n  e.  NN  |  ( 1  <  n  /\  A. m  e.  NN  (
( n  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  n ) ) ) }  C_  NN
31, 2eqsstri 3208 . 2  |-  S  C_  NN
4 infpn 12959 . . . . 5  |-  ( j  e.  NN  ->  E. k  e.  NN  ( j  < 
k  /\  A. m  e.  NN  ( ( k  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) )
5 nnge1 9772 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  1  <_  j )
65adantr 451 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  1  <_  j )
7 nnre 9753 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  j  e.  RR )
8 nnre 9753 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
9 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
10 lelttr 8912 . . . . . . . . . . . 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 2655 . . . . 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 4027 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  <  n  <->  1  <  k ) )
21 oveq1 5865 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
n  /  m )  =  ( k  /  m ) )
2221eleq1d 2349 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
( n  /  m
)  e.  NN  <->  ( k  /  m )  e.  NN ) )
23 equequ2 1649 . . . . . . . . . . . 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 2563 . . . . . . . . 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 2925 . . . . . . 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 2572 . . . 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 2608 . 2  |-  A. j  e.  NN  E. k  e.  S  j  <  k
37 unben 12956 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   class class class wbr 4023  (class class class)co 5858    ~~ cen 6860   RRcr 8736   1c1 8738    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-fac 11289
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