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Theorem ssnnssfz 28045
Description: For any finite subset of  NN, find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.)
Assertion
Ref Expression
ssnnssfz  |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  E. n  e.  NN  A  C_  (
1 ... n ) )
Distinct variable group:    A, n

Proof of Theorem ssnnssfz
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1nn 10587 . . 3  |-  1  e.  NN
2 simpr 459 . . . 4  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =  (/) )  ->  A  =  (/) )
3 0ss 3768 . . . 4  |-  (/)  C_  (
1 ... 1 )
42, 3syl6eqss 3492 . . 3  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =  (/) )  ->  A  C_  ( 1 ... 1 ) )
5 oveq2 6286 . . . . 5  |-  ( n  =  1  ->  (
1 ... n )  =  ( 1 ... 1
) )
65sseq2d 3470 . . . 4  |-  ( n  =  1  ->  ( A  C_  ( 1 ... n )  <->  A  C_  (
1 ... 1 ) ) )
76rspcev 3160 . . 3  |-  ( ( 1  e.  NN  /\  A  C_  ( 1 ... 1 ) )  ->  E. n  e.  NN  A  C_  ( 1 ... n ) )
81, 4, 7sylancr 661 . 2  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =  (/) )  ->  E. n  e.  NN  A  C_  ( 1 ... n ) )
9 elin 3626 . . . . . . 7  |-  ( A  e.  ( ~P NN  i^i  Fin )  <->  ( A  e.  ~P NN  /\  A  e.  Fin ) )
109simplbi 458 . . . . . 6  |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  A  e.  ~P NN )
1110adantr 463 . . . . 5  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  e.  ~P NN )
1211elpwid 3965 . . . 4  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  C_  NN )
13 nnssre 10580 . . . . . . 7  |-  NN  C_  RR
14 ltso 9696 . . . . . . 7  |-  <  Or  RR
15 soss 4762 . . . . . . 7  |-  ( NN  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN ) )
1613, 14, 15mp2 9 . . . . . 6  |-  <  Or  NN
1716a1i 11 . . . . 5  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  <  Or  NN )
189simprbi 462 . . . . . 6  |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  A  e.  Fin )
1918adantr 463 . . . . 5  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  e.  Fin )
20 simpr 459 . . . . 5  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
21 fisupcl 7961 . . . . 5  |-  ( (  <  Or  NN  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  NN ) )  ->  sup ( A ,  NN ,  <  )  e.  A
)
2217, 19, 20, 12, 21syl13anc 1232 . . . 4  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  sup ( A ,  NN ,  <  )  e.  A
)
2312, 22sseldd 3443 . . 3  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  sup ( A ,  NN ,  <  )  e.  NN )
2412sselda 3442 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  NN )
25 nnuz 11162 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2624, 25syl6eleq 2500 . . . . . 6  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ( ZZ>= ` 
1 ) )
2724nnzd 11007 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ZZ )
2812adantr 463 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  A  C_  NN )
2922adantr 463 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  A )
3028, 29sseldd 3443 . . . . . . . 8  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  NN )
3130nnzd 11007 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  ZZ )
32 fisup2g 7960 . . . . . . . . . . . 12  |-  ( (  <  Or  NN  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  NN ) )  ->  E. x  e.  A  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
3317, 19, 20, 12, 32syl13anc 1232 . . . . . . . . . . 11  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
34 ssrexv 3504 . . . . . . . . . . 11  |-  ( A 
C_  NN  ->  ( E. x  e.  A  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN  (
y  <  x  ->  E. z  e.  A  y  <  z ) )  ->  E. x  e.  NN  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) ) )
3512, 33, 34sylc 59 . . . . . . . . . 10  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  NN  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
3617, 35supub 7952 . . . . . . . . 9  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  -> 
( x  e.  A  ->  -.  sup ( A ,  NN ,  <  )  <  x ) )
3736imp 427 . . . . . . . 8  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  sup ( A ,  NN ,  <  )  <  x )
3824nnred 10591 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  RR )
3930nnred 10591 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  RR )
4038, 39lenltd 9763 . . . . . . . 8  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( x  <_  sup ( A ,  NN ,  <  )  <->  -.  sup ( A ,  NN ,  <  )  <  x ) )
4137, 40mpbird 232 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  <_  sup ( A ,  NN ,  <  ) )
42 eluz2 11133 . . . . . . 7  |-  ( sup ( A ,  NN ,  <  )  e.  (
ZZ>= `  x )  <->  ( x  e.  ZZ  /\  sup ( A ,  NN ,  <  )  e.  ZZ  /\  x  <_  sup ( A ,  NN ,  <  ) ) )
4327, 31, 41, 42syl3anbrc 1181 . . . . . 6  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  ( ZZ>= `  x )
)
44 eluzfz 11737 . . . . . 6  |-  ( ( x  e.  ( ZZ>= ` 
1 )  /\  sup ( A ,  NN ,  <  )  e.  ( ZZ>= `  x ) )  ->  x  e.  ( 1 ... sup ( A ,  NN ,  <  ) ) )
4526, 43, 44syl2anc 659 . . . . 5  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ( 1 ... sup ( A ,  NN ,  <  ) ) )
4645ex 432 . . . 4  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  -> 
( x  e.  A  ->  x  e.  ( 1 ... sup ( A ,  NN ,  <  ) ) ) )
4746ssrdv 3448 . . 3  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  C_  ( 1 ...
sup ( A ,  NN ,  <  ) ) )
48 oveq2 6286 . . . . 5  |-  ( n  =  sup ( A ,  NN ,  <  )  ->  ( 1 ... n )  =  ( 1 ... sup ( A ,  NN ,  <  ) ) )
4948sseq2d 3470 . . . 4  |-  ( n  =  sup ( A ,  NN ,  <  )  ->  ( A  C_  ( 1 ... n
)  <->  A  C_  ( 1 ... sup ( A ,  NN ,  <  ) ) ) )
5049rspcev 3160 . . 3  |-  ( ( sup ( A ,  NN ,  <  )  e.  NN  /\  A  C_  ( 1 ... sup ( A ,  NN ,  <  ) ) )  ->  E. n  e.  NN  A  C_  ( 1 ... n ) )
5123, 47, 50syl2anc 659 . 2  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  E. n  e.  NN  A  C_  ( 1 ... n ) )
528, 51pm2.61dane 2721 1  |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  E. n  e.  NN  A  C_  (
1 ... n ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755    i^i cin 3413    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   class class class wbr 4395    Or wor 4743   ` cfv 5569  (class class class)co 6278   Fincfn 7554   supcsup 7934   RRcr 9521   1c1 9523    < clt 9658    <_ cle 9659   NNcn 10576   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727
This theorem is referenced by:  esumfsup  28517  esumpcvgval  28525
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