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Theorem bndndx 8927
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 8925 . . . 4  |-  ( x  e.  RR  ->  E. k  e.  NN  x  <  k
)
2 nnre 8684 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  RR )
3 lelttr 7816 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  (
( A  <_  x  /\  x  <  k )  ->  A  <  k
) )
4 ltle 7815 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  k  e.  RR )  ->  ( A  <  k  ->  A  <_  k )
)
543adant2 983 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  ( A  <  k  ->  A  <_  k ) )
63, 5syld 45 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  RR )  ->  (
( A  <_  x  /\  x  <  k )  ->  A  <_  k
) )
76exp5o 1187 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
x  e.  RR  ->  ( k  e.  RR  ->  ( A  <_  x  ->  ( x  <  k  ->  A  <_  k ) ) ) ) )
87com3l 81 . . . . . . . 8  |-  ( x  e.  RR  ->  (
k  e.  RR  ->  ( A  e.  RR  ->  ( A  <_  x  ->  ( x  <  k  ->  A  <_  k ) ) ) ) )
98imp4b 345 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  (
x  <  k  ->  A  <_  k ) ) )
109com23 78 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( x  <  k  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  A  <_  k ) ) )
112, 10sylan2 282 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( x  <  k  ->  ( ( A  e.  RR  /\  A  <_  x )  ->  A  <_  k ) ) )
1211reximdva 2509 . . . 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 2556 . . 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 2518 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 103    /\ w3a 945    e. wcel 1463   A.wral 2391   E.wrex 2392   class class class wbr 3897   RRcr 7583    < clt 7764    <_ cle 7765   NNcn 8677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1re 7678  ax-addrcl 7681  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-arch 7703
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-inn 8678
This theorem is referenced by: (None)
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