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Theorem bndndx 8354
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 8352 . . . 4  |-  ( x  e.  RR  ->  E. k  e.  NN  x  <  k
)
2 nnre 8113 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  RR )
3 lelttr 7266 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  (
( A  <_  x  /\  x  <  k )  ->  A  <  k
) )
4 ltle 7265 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  k  e.  RR )  ->  ( A  <  k  ->  A  <_  k )
)
543adant2 958 . . . . . . . . . . 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 1158 . . . . . . . . 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 2464 . . . 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 2505 . . 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 2472 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 920    e. wcel 1434   A.wral 2349   E.wrex 2350   class class class wbr 3793   RRcr 7042    < clt 7215    <_ cle 7216   NNcn 8106
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1re 7132  ax-addrcl 7135  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-arch 7157
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-inn 8107
This theorem is referenced by: (None)
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