Theorem bndndx 9368

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 9366	. . . 4 |- ( x e. RR -> E. k e. NN x < k )
2		nnre 9117	. . . . . 6 |- ( k e. NN -> k e. RR )
3		lelttr 8235	. . . . . . . . . . 11 |- ( ( A e. RR /\ x e. RR /\ k e. RR ) -> ( ( A <_ x /\ x < k ) -> A < k ) )
4		ltle 8234	. . . . . . . . . . . 12 |- ( ( A e. RR /\ k e. RR ) -> ( A < k -> A <_ k ) )
5	4	3adant2 1040	. . . . . . . . . . 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 1250	. . . . . . . . 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 2632	. . . 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 2681	. . 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 2642	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 1002   e. wcel 2200   A.wral 2508   E.wrex 2509   class class class wbr 4083   RRcr 7998   < clt 8181   <_ cle 8182   NNcn 9110

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-arch 8118

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-inn 9111

This theorem is referenced by: (None)