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Theorem infpn2 12922
Description: There exist infinitely many prime numbers: the set of all primes  S is unbounded by infpn 12921, so by unben 12918 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
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 3233 . . 3  |-  { n  e.  NN  |  ( 1  <  n  /\  A. m  e.  NN  (
( n  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  n ) ) ) }  C_  NN
31, 2eqsstri 3183 . 2  |-  S  C_  NN
4 infpn 12921 . . . . 5  |-  ( j  e.  NN  ->  E. k  e.  NN  ( j  < 
k  /\  A. m  e.  NN  ( ( k  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) )
5 nnge1 9740 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  1  <_  j )
65adantr 453 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  1  <_  j )
7 nnre 9721 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  j  e.  RR )
8 nnre 9721 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
9 1re 8805 . . . . . . . . . . . 12  |-  1  e.  RR
10 lelttr 8880 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  j  e.  RR  /\  k  e.  RR )  ->  (
( 1  <_  j  /\  j  <  k )  ->  1  <  k
) )
119, 10mp3an1 1269 . . . . . . . . . . 11  |-  ( ( j  e.  RR  /\  k  e.  RR )  ->  ( ( 1  <_ 
j  /\  j  <  k )  ->  1  <  k ) )
127, 8, 11syl2an 465 . . . . . . . . . 10  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( ( 1  <_ 
j  /\  j  <  k )  ->  1  <  k ) )
136, 12mpand 659 . . . . . . . . 9  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( j  <  k  ->  1  <  k ) )
1413ancld 538 . . . . . . . 8  |-  ( ( j  e.  NN  /\  k  e.  NN )  ->  ( j  <  k  ->  ( j  <  k  /\  1  <  k ) ) )
1514anim1d 549 . . . . . . 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 633 . . . . . . 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 219 . . . . . 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 2630 . . . . 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 16 . . . 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 4001 . . . . . . . . 9  |-  ( n  =  k  ->  (
1  <  n  <->  1  <  k ) )
21 oveq1 5799 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
n  /  m )  =  ( k  /  m ) )
2221eleq1d 2324 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
( n  /  m
)  e.  NN  <->  ( k  /  m )  e.  NN ) )
23 equequ2 1830 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
m  =  n  <->  m  =  k ) )
2423orbi2d 685 . . . . . . . . . . 11  |-  ( n  =  k  ->  (
( m  =  1  \/  m  =  n )  <->  ( m  =  1  \/  m  =  k ) ) )
2522, 24imbi12d 313 . . . . . . . . . 10  |-  ( n  =  k  ->  (
( ( n  /  m )  e.  NN  ->  ( m  =  1  \/  m  =  n ) )  <->  ( (
k  /  m )  e.  NN  ->  (
m  =  1  \/  m  =  k ) ) ) )
2625ralbidv 2538 . . . . . . . . 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 694 . . . . . . . 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 2900 . . . . . . 7  |-  ( k  e.  S  <->  ( k  e.  NN  /\  ( 1  <  k  /\  A. m  e.  NN  (
( k  /  m
)  e.  NN  ->  ( m  =  1  \/  m  =  k ) ) ) ) )
2928anbi1i 679 . . . . . 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 633 . . . . . 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 439 . . . . . . 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 678 . . . . . 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 264 . . . . 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 2547 . . . 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 205 . . 3  |-  ( j  e.  NN  ->  E. k  e.  S  j  <  k )
3635rgen 2583 . 2  |-  A. j  e.  NN  E. k  e.  S  j  <  k
37 unben 12918 . 2  |-  ( ( S  C_  NN  /\  A. j  e.  NN  E. k  e.  S  j  <  k )  ->  S  ~~  NN )
383, 36, 37mp2an 656 1  |-  S  ~~  NN
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518   E.wrex 2519   {crab 2522    C_ wss 3127   class class class wbr 3997  (class class class)co 5792    ~~ cen 6828   RRcr 8704   1c1 8706    < clt 8835    <_ cle 8836    / cdiv 9391   NNcn 9714
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-n0 9933  df-z 9992  df-uz 10198  df-seq 11013  df-fac 11255
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