Theorem bndndx 9483

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 9481	. . . 4 |- ( x e. RR -> E. k e. NN x < k )
2		nnre 9232	. . . . . 6 |- ( k e. NN -> k e. RR )
3		lelttr 8350	. . . . . . . . . . 11 |- ( ( A e. RR /\ x e. RR /\ k e. RR ) -> ( ( A <_ x /\ x < k ) -> A < k ) )
4		ltle 8349	. . . . . . . . . . . 12 |- ( ( A e. RR /\ k e. RR ) -> ( A < k -> A <_ k ) )
5	4	3adant2 1043	. . . . . . . . . . 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 1253	. . . . . . . . 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 2644	. . . 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 2693	. . 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 2654	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 1005   e. wcel 2203   A.wral 2520   E.wrex 2521   class class class wbr 4102   RRcr 8114   < clt 8296   <_ cle 8297   NNcn 9225

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-cnex 8206  ax-resscn 8207  ax-1re 8209  ax-addrcl 8212  ax-pre-ltirr 8227  ax-pre-ltwlin 8228  ax-pre-lttrn 8229  ax-arch 8234

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-int 3943  df-br 4103  df-opab 4165  df-xp 4746  df-cnv 4748  df-pnf 8298  df-mnf 8299  df-xr 8300  df-ltxr 8301  df-le 8302  df-inn 9226

This theorem is referenced by: (None)