Theorem squeeze0 8879

Description: If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Assertion

Ref	Expression
squeeze0	|- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A = 0 )

Distinct variable group:   x, A

Proof of Theorem squeeze0

Step	Hyp	Ref	Expression
1		ltnr 8052	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	3ad2ant1 1020	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. A < A )
3		breq2 4022	. . . . . . 7 |- ( x = A -> ( 0 < x <-> 0 < A ) )
4		breq2 4022	. . . . . . 7 |- ( x = A -> ( A < x <-> A < A ) )
5	3, 4	imbi12d 234	. . . . . 6 |- ( x = A -> ( ( 0 < x -> A < x ) <-> ( 0 < A -> A < A ) ) )
6	5	rspcva 2854	. . . . 5 |- ( ( A e. RR /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
7	6	3adant2 1018	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
8	2, 7	mtod 664	. . 3 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. 0 < A )
9		simp1 999	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A e. RR )
10		0red 7976	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 e. RR )
11	9, 10	lenltd 8093	. . 3 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( A <_ 0 <-> -. 0 < A ) )
12	8, 11	mpbird 167	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A <_ 0 )
13		simp2 1000	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 <_ A )
14	9, 10	letri3d 8091	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( A = 0 <-> ( A <_ 0 /\ 0 <_ A ) ) )
15	12, 13, 14	mpbir2and 946	1 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A = 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   class class class wbr 4018   RRcr 7828   0cc0 7829   < clt 8010   <_ cle 8011

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926  ax-rnegex 7938  ax-pre-ltirr 7941  ax-pre-apti 7944

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-cnv 4649  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016

This theorem is referenced by: (None)