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Theorem ssnnssfz 27265
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 10543 . . 3  |-  1  e.  NN
2 simpr 461 . . . 4  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =  (/) )  ->  A  =  (/) )
3 0ss 3814 . . . 4  |-  (/)  C_  (
1 ... 1 )
42, 3syl6eqss 3554 . . 3  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =  (/) )  ->  A  C_  ( 1 ... 1 ) )
5 oveq2 6290 . . . . 5  |-  ( n  =  1  ->  (
1 ... n )  =  ( 1 ... 1
) )
65sseq2d 3532 . . . 4  |-  ( n  =  1  ->  ( A  C_  ( 1 ... n )  <->  A  C_  (
1 ... 1 ) ) )
76rspcev 3214 . . 3  |-  ( ( 1  e.  NN  /\  A  C_  ( 1 ... 1 ) )  ->  E. n  e.  NN  A  C_  ( 1 ... n ) )
81, 4, 7sylancr 663 . 2  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =  (/) )  ->  E. n  e.  NN  A  C_  ( 1 ... n ) )
9 elin 3687 . . . . . . 7  |-  ( A  e.  ( ~P NN  i^i  Fin )  <->  ( A  e.  ~P NN  /\  A  e.  Fin ) )
109simplbi 460 . . . . . 6  |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  A  e.  ~P NN )
1110adantr 465 . . . . 5  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  e.  ~P NN )
1211elpwid 4020 . . . 4  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  C_  NN )
13 nnssre 10536 . . . . . . 7  |-  NN  C_  RR
14 ltso 9661 . . . . . . 7  |-  <  Or  RR
15 soss 4818 . . . . . . 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 464 . . . . . 6  |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  A  e.  Fin )
1918adantr 465 . . . . 5  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  e.  Fin )
20 simpr 461 . . . . 5  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
21 fisupcl 7923 . . . . 5  |-  ( (  <  Or  NN  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_  NN ) )  ->  sup ( A ,  NN ,  <  )  e.  A
)
2217, 19, 20, 12, 21syl13anc 1230 . . . 4  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  sup ( A ,  NN ,  <  )  e.  A
)
2312, 22sseldd 3505 . . 3  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  sup ( A ,  NN ,  <  )  e.  NN )
2412sselda 3504 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  NN )
25 nnuz 11113 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2624, 25syl6eleq 2565 . . . . . 6  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ( ZZ>= ` 
1 ) )
2724nnzd 10961 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ZZ )
2812adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  A  C_  NN )
2922adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  A )
3028, 29sseldd 3505 . . . . . . . 8  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  NN )
3130nnzd 10961 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  ZZ )
32 fisup2g 7922 . . . . . . . . . . . 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 1230 . . . . . . . . . . 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 3565 . . . . . . . . . . 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 60 . . . . . . . . . 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 7915 . . . . . . . . 9  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  -> 
( x  e.  A  ->  -.  sup ( A ,  NN ,  <  )  <  x ) )
3736imp 429 . . . . . . . 8  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  sup ( A ,  NN ,  <  )  <  x )
3824nnred 10547 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  RR )
3930nnred 10547 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  RR )
4038, 39lenltd 9726 . . . . . . . 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 11084 . . . . . . 7  |-  ( sup ( A ,  NN ,  <  )  e.  (
ZZ>= `  x )  <->  ( x  e.  ZZ  /\  sup ( A ,  NN ,  <  )  e.  ZZ  /\  x  <_  sup ( A ,  NN ,  <  ) ) )
4327, 31, 41, 42syl3anbrc 1180 . . . . . 6  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN ,  <  )  e.  ( ZZ>= `  x )
)
44 eluzfz 11679 . . . . . 6  |-  ( ( x  e.  ( ZZ>= ` 
1 )  /\  sup ( A ,  NN ,  <  )  e.  ( ZZ>= `  x ) )  ->  x  e.  ( 1 ... sup ( A ,  NN ,  <  ) ) )
4526, 43, 44syl2anc 661 . . . . 5  |-  ( ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ( 1 ... sup ( A ,  NN ,  <  ) ) )
4645ex 434 . . . 4  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  -> 
( x  e.  A  ->  x  e.  ( 1 ... sup ( A ,  NN ,  <  ) ) ) )
4746ssrdv 3510 . . 3  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  A  C_  ( 1 ...
sup ( A ,  NN ,  <  ) ) )
48 oveq2 6290 . . . . 5  |-  ( n  =  sup ( A ,  NN ,  <  )  ->  ( 1 ... n )  =  ( 1 ... sup ( A ,  NN ,  <  ) ) )
4948sseq2d 3532 . . . 4  |-  ( n  =  sup ( A ,  NN ,  <  )  ->  ( A  C_  ( 1 ... n
)  <->  A  C_  ( 1 ... sup ( A ,  NN ,  <  ) ) ) )
5049rspcev 3214 . . 3  |-  ( ( sup ( A ,  NN ,  <  )  e.  NN  /\  A  C_  ( 1 ... sup ( A ,  NN ,  <  ) ) )  ->  E. n  e.  NN  A  C_  ( 1 ... n ) )
5123, 47, 50syl2anc 661 . 2  |-  ( ( A  e.  ( ~P NN  i^i  Fin )  /\  A  =/=  (/) )  ->  E. n  e.  NN  A  C_  ( 1 ... n ) )
528, 51pm2.61dane 2785 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 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447    Or wor 4799   ` cfv 5586  (class class class)co 6282   Fincfn 7513   supcsup 7896   RRcr 9487   1c1 9489    < clt 9624    <_ cle 9625   NNcn 10532   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669
This theorem is referenced by:  esumfsup  27716  esumpcvgval  27724
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