Theorem bndndx 9544

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

Step	Hyp	Ref	Expression
1		arch 9542	. . . 4 |- ( x e. RR -> E. k e. NN x < k )
2		nnre 9293	. . . . . 6 |- ( k e. NN -> k e. RR )
3		lelttr 8407	. . . . . . . . . . 11 |- ( ( A e. RR /\ x e. RR /\ k e. RR ) -> ( ( A <_ x /\ x < k ) -> A < k ) )
4		ltle 8406	. . . . . . . . . . . 12 |- ( ( A e. RR /\ k e. RR ) -> ( A < k -> A <_ k ) )
5	4	3adant2 1047	. . . . . . . . . . 11 |- ( ( A e. RR /\ x e. RR /\ k e. RR ) -> ( A < k -> A <_ k ) )
6	3, 5	syld 45	. . . . . . . . . 10 |- ( ( A e. RR /\ x e. RR /\ k e. RR ) -> ( ( A <_ x /\ x < k ) -> A <_ k ) )
7	6	exp5o 1257	. . . . . . . . 9 |- ( A e. RR -> ( x e. RR -> ( k e. RR -> ( A <_ x -> ( x < k -> A <_ k ) ) ) ) )
8	7	com3l 81	. . . . . . . 8 |- ( x e. RR -> ( k e. RR -> ( A e. RR -> ( A <_ x -> ( x < k -> A <_ k ) ) ) ) )
9	8	imp4b 350	. . . . . . 7 |- ( ( x e. RR /\ k e. RR ) -> ( ( A e. RR /\ A <_ x ) -> ( x < k -> A <_ k ) ) )
10	9	com23 78	. . . . . 6 |- ( ( x e. RR /\ k e. RR ) -> ( x < k -> ( ( A e. RR /\ A <_ x ) -> A <_ k ) ) )
11	2, 10	sylan2 286	. . . . 5 |- ( ( x e. RR /\ k e. NN ) -> ( x < k -> ( ( A e. RR /\ A <_ x ) -> A <_ k ) ) )
12	11	reximdva 2652	. . . 4 |- ( x e. RR -> ( E. k e. NN x < k -> E. k e. NN ( ( A e. RR /\ A <_ x ) -> A <_ k ) ) )
13	1, 12	mpd 13	. . 3 |- ( x e. RR -> E. k e. NN ( ( A e. RR /\ A <_ x ) -> A <_ k ) )
14		r19.35-1 2701	. . 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 ) )
15	13, 14	syl 14	. 2 |- ( x e. RR -> ( A. k e. NN ( A e. RR /\ A <_ x ) -> E. k e. NN A <_ k ) )
16	15	rexlimiv 2662	1 |- ( E. x e. RR A. k e. NN ( A e. RR /\ A <_ x ) -> E. k e. NN A <_ k )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   e. wcel 2209   A.wral 2528   E.wrex 2529   class class class wbr 4128   RRcr 8171   < clt 8353   <_ cle 8354   NNcn 9286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1re 8266  ax-addrcl 8269  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-arch 8291

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-xp 4778  df-cnv 4780  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-inn 9287

This theorem is referenced by: (None)