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Theorem uzfissfz 37549
Description: For any finite subset of the upper integers, there is a finite set of sequential integers that includes it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
uzfissfz.m  |-  ( ph  ->  M  e.  ZZ )
uzfissfz.z  |-  Z  =  ( ZZ>= `  M )
uzfissfz.a  |-  ( ph  ->  A  C_  Z )
uzfissfz.fi  |-  ( ph  ->  A  e.  Fin )
Assertion
Ref Expression
uzfissfz  |-  ( ph  ->  E. k  e.  Z  A  C_  ( M ... k ) )
Distinct variable groups:    A, k    k, M    k, Z
Allowed substitution hint:    ph( k)

Proof of Theorem uzfissfz
Dummy variables  j  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzfissfz.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2 uzid 11173 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 17 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 uzfissfz.z . . . . . . 7  |-  Z  =  ( ZZ>= `  M )
54a1i 11 . . . . . 6  |-  ( ph  ->  Z  =  ( ZZ>= `  M ) )
65eqcomd 2457 . . . . 5  |-  ( ph  ->  ( ZZ>= `  M )  =  Z )
73, 6eleqtrd 2531 . . . 4  |-  ( ph  ->  M  e.  Z )
87adantr 467 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  M  e.  Z )
9 id 22 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
10 0ss 3763 . . . . . 6  |-  (/)  C_  ( M ... M )
1110a1i 11 . . . . 5  |-  ( A  =  (/)  ->  (/)  C_  ( M ... M ) )
129, 11eqsstrd 3466 . . . 4  |-  ( A  =  (/)  ->  A  C_  ( M ... M ) )
1312adantl 468 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  A  C_  ( M ... M ) )
14 oveq2 6298 . . . . 5  |-  ( k  =  M  ->  ( M ... k )  =  ( M ... M
) )
1514sseq2d 3460 . . . 4  |-  ( k  =  M  ->  ( A  C_  ( M ... k )  <->  A  C_  ( M ... M ) ) )
1615rspcev 3150 . . 3  |-  ( ( M  e.  Z  /\  A  C_  ( M ... M ) )  ->  E. k  e.  Z  A  C_  ( M ... k ) )
178, 13, 16syl2anc 667 . 2  |-  ( (
ph  /\  A  =  (/) )  ->  E. k  e.  Z  A  C_  ( M ... k ) )
18 uzfissfz.a . . . . 5  |-  ( ph  ->  A  C_  Z )
1918adantr 467 . . . 4  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  C_  Z )
20 uzssz 11178 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  ZZ
214, 20eqsstri 3462 . . . . . . . 8  |-  Z  C_  ZZ
2221a1i 11 . . . . . . 7  |-  ( ph  ->  Z  C_  ZZ )
2318, 22sstrd 3442 . . . . . 6  |-  ( ph  ->  A  C_  ZZ )
2423adantr 467 . . . . 5  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  C_  ZZ )
259necon3bi 2650 . . . . . 6  |-  ( -.  A  =  (/)  ->  A  =/=  (/) )
2625adantl 468 . . . . 5  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  =/=  (/) )
27 uzfissfz.fi . . . . . 6  |-  ( ph  ->  A  e.  Fin )
2827adantr 467 . . . . 5  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  e.  Fin )
29 suprfinzcl 11050 . . . . 5  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  sup ( A ,  RR ,  <  )  e.  A )
3024, 26, 28, 29syl3anc 1268 . . . 4  |-  ( (
ph  /\  -.  A  =  (/) )  ->  sup ( A ,  RR ,  <  )  e.  A )
3119, 30sseldd 3433 . . 3  |-  ( (
ph  /\  -.  A  =  (/) )  ->  sup ( A ,  RR ,  <  )  e.  Z )
321ad2antrr 732 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  M  e.  ZZ )
3321, 31sseldi 3430 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  =  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
3433adantr 467 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
3524sselda 3432 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  j  e.  ZZ )
3632, 34, 353jca 1188 . . . . . . 7  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  ( M  e.  ZZ  /\ 
sup ( A ,  RR ,  <  )  e.  ZZ  /\  j  e.  ZZ ) )
3718sselda 3432 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  Z )
384a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  Z  =  ( ZZ>= `  M
) )
3937, 38eleqtrd 2531 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  ( ZZ>= `  M )
)
40 eluzle 11171 . . . . . . . . 9  |-  ( j  e.  ( ZZ>= `  M
)  ->  M  <_  j )
4139, 40syl 17 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  M  <_  j )
4241adantlr 721 . . . . . . 7  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  M  <_  j )
43 zssre 10944 . . . . . . . . . 10  |-  ZZ  C_  RR
4423, 43syl6ss 3444 . . . . . . . . 9  |-  ( ph  ->  A  C_  RR )
4544ad2antrr 732 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  A  C_  RR )
4626adantr 467 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  A  =/=  (/) )
47 fimaxre2 10552 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
4844, 27, 47syl2anc 667 . . . . . . . . 9  |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )
4948ad2antrr 732 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )
50 simpr 463 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  j  e.  A )
51 suprub 10570 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  j  e.  A )  ->  j  <_  sup ( A ,  RR ,  <  ) )
5245, 46, 49, 50, 51syl31anc 1271 . . . . . . 7  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  j  <_  sup ( A ,  RR ,  <  ) )
5336, 42, 52jca32 538 . . . . . 6  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  ( ( M  e.  ZZ  /\  sup ( A ,  RR ,  <  )  e.  ZZ  /\  j  e.  ZZ )  /\  ( M  <_  j  /\  j  <_  sup ( A ,  RR ,  <  ) ) ) )
54 elfz2 11791 . . . . . 6  |-  ( j  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  ( ( M  e.  ZZ  /\  sup ( A ,  RR ,  <  )  e.  ZZ  /\  j  e.  ZZ )  /\  ( M  <_  j  /\  j  <_  sup ( A ,  RR ,  <  ) ) ) )
5553, 54sylibr 216 . . . . 5  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  j  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
5655ralrimiva 2802 . . . 4  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A. j  e.  A  j  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
57 dfss3 3422 . . . 4  |-  ( A 
C_  ( M ... sup ( A ,  RR ,  <  ) )  <->  A. j  e.  A  j  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
5856, 57sylibr 216 . . 3  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
59 oveq2 6298 . . . . 5  |-  ( k  =  sup ( A ,  RR ,  <  )  ->  ( M ... k )  =  ( M ... sup ( A ,  RR ,  <  ) ) )
6059sseq2d 3460 . . . 4  |-  ( k  =  sup ( A ,  RR ,  <  )  ->  ( A  C_  ( M ... k )  <-> 
A  C_  ( M ... sup ( A ,  RR ,  <  ) ) ) )
6160rspcev 3150 . . 3  |-  ( ( sup ( A ,  RR ,  <  )  e.  Z  /\  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )  ->  E. k  e.  Z  A  C_  ( M ... k ) )
6231, 58, 61syl2anc 667 . 2  |-  ( (
ph  /\  -.  A  =  (/) )  ->  E. k  e.  Z  A  C_  ( M ... k ) )
6317, 62pm2.61dan 800 1  |-  ( ph  ->  E. k  e.  Z  A  C_  ( M ... k ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    C_ wss 3404   (/)c0 3731   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   Fincfn 7569   supcsup 7954   RRcr 9538    < clt 9675    <_ cle 9676   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785
This theorem is referenced by:  sge0uzfsumgt  38286  carageniuncllem2  38343  caratheodorylem2  38348
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