Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssnn0ssfz Structured version   Unicode version

Theorem ssnn0ssfz 38902
Description: For any finite subset of  NN0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 28203. (Contributed by AV, 30-Sep-2019.)
Assertion
Ref Expression
ssnn0ssfz  |-  ( A  e.  ( ~P NN0  i^i 
Fin )  ->  E. n  e.  NN0  A  C_  (
0 ... n ) )
Distinct variable group:    A, n

Proof of Theorem ssnn0ssfz
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0nn0 10884 . . 3  |-  0  e.  NN0
2 simpr 462 . . . 4  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =  (/) )  ->  A  =  (/) )
3 0ss 3797 . . . 4  |-  (/)  C_  (
0 ... 0 )
42, 3syl6eqss 3520 . . 3  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =  (/) )  ->  A  C_  ( 0 ... 0 ) )
5 oveq2 6313 . . . . 5  |-  ( n  =  0  ->  (
0 ... n )  =  ( 0 ... 0
) )
65sseq2d 3498 . . . 4  |-  ( n  =  0  ->  ( A  C_  ( 0 ... n )  <->  A  C_  (
0 ... 0 ) ) )
76rspcev 3188 . . 3  |-  ( ( 0  e.  NN0  /\  A  C_  ( 0 ... 0 ) )  ->  E. n  e.  NN0  A 
C_  ( 0 ... n ) )
81, 4, 7sylancr 667 . 2  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =  (/) )  ->  E. n  e.  NN0  A 
C_  ( 0 ... n ) )
9 elin 3655 . . . . . . 7  |-  ( A  e.  ( ~P NN0  i^i 
Fin )  <->  ( A  e.  ~P NN0  /\  A  e.  Fin ) )
109simplbi 461 . . . . . 6  |-  ( A  e.  ( ~P NN0  i^i 
Fin )  ->  A  e.  ~P NN0 )
1110adantr 466 . . . . 5  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  A  e.  ~P NN0 )
1211elpwid 3995 . . . 4  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  A  C_  NN0 )
13 nn0ssre 10873 . . . . . . 7  |-  NN0  C_  RR
14 ltso 9713 . . . . . . 7  |-  <  Or  RR
15 soss 4793 . . . . . . 7  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
1613, 14, 15mp2 9 . . . . . 6  |-  <  Or  NN0
1716a1i 11 . . . . 5  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  <  Or  NN0 )
189simprbi 465 . . . . . 6  |-  ( A  e.  ( ~P NN0  i^i 
Fin )  ->  A  e.  Fin )
1918adantr 466 . . . . 5  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  A  e.  Fin )
20 simpr 462 . . . . 5  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
21 fisupcl 7991 . . . . 5  |-  ( (  <  Or  NN0  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_ 
NN0 ) )  ->  sup ( A ,  NN0 ,  <  )  e.  A
)
2217, 19, 20, 12, 21syl13anc 1266 . . . 4  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  sup ( A ,  NN0 ,  <  )  e.  A
)
2312, 22sseldd 3471 . . 3  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  sup ( A ,  NN0 ,  <  )  e.  NN0 )
2412sselda 3470 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  NN0 )
25 nn0uz 11193 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2624, 25syl6eleq 2527 . . . . . 6  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ( ZZ>= ` 
0 ) )
2724nn0zd 11038 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ZZ )
2812adantr 466 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  A  C_  NN0 )
2922adantr 466 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN0 ,  <  )  e.  A )
3028, 29sseldd 3471 . . . . . . . 8  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN0 ,  <  )  e. 
NN0 )
3130nn0zd 11038 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN0 ,  <  )  e.  ZZ )
32 fisup2g 7990 . . . . . . . . . . . 12  |-  ( (  <  Or  NN0  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A  C_ 
NN0 ) )  ->  E. x  e.  A  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN0  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
3317, 19, 20, 12, 32syl13anc 1266 . . . . . . . . . . 11  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN0  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
34 ssrexv 3532 . . . . . . . . . . 11  |-  ( A 
C_  NN0  ->  ( E. x  e.  A  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN0  (
y  <  x  ->  E. z  e.  A  y  <  z ) )  ->  E. x  e.  NN0  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN0  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) ) )
3512, 33, 34sylc 62 . . . . . . . . . 10  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  NN0  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  NN0  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
3617, 35supub 7979 . . . . . . . . 9  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  -> 
( x  e.  A  ->  -.  sup ( A ,  NN0 ,  <  )  <  x ) )
3736imp 430 . . . . . . . 8  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  sup ( A ,  NN0 ,  <  )  <  x )
3824nn0red 10926 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  RR )
3930nn0red 10926 . . . . . . . . 9  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN0 ,  <  )  e.  RR )
4038, 39lenltd 9780 . . . . . . . 8  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( x  <_  sup ( A ,  NN0 ,  <  )  <->  -.  sup ( A ,  NN0 ,  <  )  <  x ) )
4137, 40mpbird 235 . . . . . . 7  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  <_  sup ( A ,  NN0 ,  <  ) )
42 eluz2 11165 . . . . . . 7  |-  ( sup ( A ,  NN0 ,  <  )  e.  (
ZZ>= `  x )  <->  ( x  e.  ZZ  /\  sup ( A ,  NN0 ,  <  )  e.  ZZ  /\  x  <_  sup ( A ,  NN0 ,  <  ) ) )
4327, 31, 41, 42syl3anbrc 1189 . . . . . 6  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  sup ( A ,  NN0 ,  <  )  e.  ( ZZ>= `  x )
)
44 eluzfz 11793 . . . . . 6  |-  ( ( x  e.  ( ZZ>= ` 
0 )  /\  sup ( A ,  NN0 ,  <  )  e.  ( ZZ>= `  x ) )  ->  x  e.  ( 0 ... sup ( A ,  NN0 ,  <  ) ) )
4526, 43, 44syl2anc 665 . . . . 5  |-  ( ( ( A  e.  ( ~P NN0  i^i  Fin )  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  ( 0 ... sup ( A ,  NN0 ,  <  ) ) )
4645ex 435 . . . 4  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  -> 
( x  e.  A  ->  x  e.  ( 0 ... sup ( A ,  NN0 ,  <  ) ) ) )
4746ssrdv 3476 . . 3  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  A  C_  ( 0 ...
sup ( A ,  NN0 ,  <  ) ) )
48 oveq2 6313 . . . . 5  |-  ( n  =  sup ( A ,  NN0 ,  <  )  ->  ( 0 ... n )  =  ( 0 ... sup ( A ,  NN0 ,  <  ) ) )
4948sseq2d 3498 . . . 4  |-  ( n  =  sup ( A ,  NN0 ,  <  )  ->  ( A  C_  ( 0 ... n
)  <->  A  C_  ( 0 ... sup ( A ,  NN0 ,  <  ) ) ) )
5049rspcev 3188 . . 3  |-  ( ( sup ( A ,  NN0 ,  <  )  e. 
NN0  /\  A  C_  (
0 ... sup ( A ,  NN0 ,  <  ) ) )  ->  E. n  e.  NN0  A  C_  (
0 ... n ) )
5123, 47, 50syl2anc 665 . 2  |-  ( ( A  e.  ( ~P
NN0  i^i  Fin )  /\  A  =/=  (/) )  ->  E. n  e.  NN0  A 
C_  ( 0 ... n ) )
528, 51pm2.61dane 2749 1  |-  ( A  e.  ( ~P NN0  i^i 
Fin )  ->  E. n  e.  NN0  A  C_  (
0 ... n ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   class class class wbr 4426    Or wor 4774   ` cfv 5601  (class class class)co 6305   Fincfn 7577   supcsup 7960   RRcr 9537   0cc0 9538    < clt 9674    <_ cle 9675   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator