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Theorem bndndx 8180
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 8178 . . . 4  |-  ( x  e.  RR  ->  E. k  e.  NN  x  <  k
)
2 nnre 7921 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  RR )
3 lelttr 7106 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  (
( A  <_  x  /\  x  <  k )  ->  A  <  k
) )
4 ltle 7105 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  k  e.  RR )  ->  ( A  <  k  ->  A  <_  k )
)
543adant2 923 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  ( A  <  k  ->  A  <_  k ) )
63, 5syld 40 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  (
( A  <_  x  /\  x  <  k )  ->  A  <_  k
) )
76exp5o 1123 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
x  e.  RR  ->  ( k  e.  RR  ->  ( A  <_  x  ->  ( x  <  k  ->  A  <_  k ) ) ) ) )
87com3l 75 . . . . . . . 8  |-  ( x  e.  RR  ->  (
k  e.  RR  ->  ( A  e.  RR  ->  ( A  <_  x  ->  ( x  <  k  ->  A  <_  k ) ) ) ) )
98imp4b 332 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  (
x  <  k  ->  A  <_  k ) ) )
109com23 72 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( x  <  k  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  A  <_  k ) ) )
112, 10sylan2 270 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( x  <  k  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  A  <_  k ) ) )
1211reximdva 2421 . . . 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 2460 . . 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 2427 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 97    /\ w3a 885    e. wcel 1393   A.wral 2306   E.wrex 2307   class class class wbr 3764   RRcr 6888    < clt 7060    <_ cle 7061   NNcn 7914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1re 6978  ax-addrcl 6981  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-arch 7003
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-inn 7915
This theorem is referenced by: (None)
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