ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bndndx Unicode version

Theorem bndndx 8642
Description: A bounded real sequence  A (
k ) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
Assertion
Ref Expression
bndndx  |-  ( E. x  e.  RR  A. k  e.  NN  ( A  e.  RR  /\  A  <_  x )  ->  E. k  e.  NN  A  <_  k
)
Distinct variable groups:    x, A    x, k
Allowed substitution hint:    A( k)

Proof of Theorem bndndx
StepHypRef Expression
1 arch 8640 . . . 4  |-  ( x  e.  RR  ->  E. k  e.  NN  x  <  k
)
2 nnre 8401 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  RR )
3 lelttr 7552 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  (
( A  <_  x  /\  x  <  k )  ->  A  <  k
) )
4 ltle 7551 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  k  e.  RR )  ->  ( A  <  k  ->  A  <_  k )
)
543adant2 962 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  ( A  <  k  ->  A  <_  k ) )
63, 5syld 44 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  (
( A  <_  x  /\  x  <  k )  ->  A  <_  k
) )
76exp5o 1162 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
x  e.  RR  ->  ( k  e.  RR  ->  ( A  <_  x  ->  ( x  <  k  ->  A  <_  k ) ) ) ) )
87com3l 80 . . . . . . . 8  |-  ( x  e.  RR  ->  (
k  e.  RR  ->  ( A  e.  RR  ->  ( A  <_  x  ->  ( x  <  k  ->  A  <_  k ) ) ) ) )
98imp4b 342 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  (
x  <  k  ->  A  <_  k ) ) )
109com23 77 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( x  <  k  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  A  <_  k ) ) )
112, 10sylan2 280 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( x  <  k  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  A  <_  k ) ) )
1211reximdva 2475 . . . 4  |-  ( x  e.  RR  ->  ( E. k  e.  NN  x  <  k  ->  E. k  e.  NN  ( ( A  e.  RR  /\  A  <_  x )  ->  A  <_  k ) ) )
131, 12mpd 13 . . 3  |-  ( x  e.  RR  ->  E. k  e.  NN  ( ( A  e.  RR  /\  A  <_  x )  ->  A  <_  k ) )
14 r19.35-1 2517 . . 3  |-  ( E. k  e.  NN  (
( A  e.  RR  /\  A  <_  x )  ->  A  <_  k )  ->  ( A. k  e.  NN  ( A  e.  RR  /\  A  <_  x )  ->  E. k  e.  NN  A  <_  k
) )
1513, 14syl 14 . 2  |-  ( x  e.  RR  ->  ( A. k  e.  NN  ( A  e.  RR  /\  A  <_  x )  ->  E. k  e.  NN  A  <_  k ) )
1615rexlimiv 2483 1  |-  ( E. x  e.  RR  A. k  e.  NN  ( A  e.  RR  /\  A  <_  x )  ->  E. k  e.  NN  A  <_  k
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    e. wcel 1438   A.wral 2359   E.wrex 2360   class class class wbr 3837   RRcr 7328    < clt 7501    <_ cle 7502   NNcn 8394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1re 7418  ax-addrcl 7421  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-inn 8395
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator