Theorem squeeze0 8686

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 7865	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	3ad2ant1 1003	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. A < A )
3		breq2 3941	. . . . . . 7 |- ( x = A -> ( 0 < x <-> 0 < A ) )
4		breq2 3941	. . . . . . 7 |- ( x = A -> ( A < x <-> A < A ) )
5	3, 4	imbi12d 233	. . . . . 6 |- ( x = A -> ( ( 0 < x -> A < x ) <-> ( 0 < A -> A < A ) ) )
6	5	rspcva 2791	. . . . 5 |- ( ( A e. RR /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
7	6	3adant2 1001	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
8	2, 7	mtod 653	. . 3 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. 0 < A )
9		simp1 982	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A e. RR )
10		0red 7791	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 e. RR )
11	9, 10	lenltd 7904	. . 3 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( A <_ 0 <-> -. 0 < A ) )
12	8, 11	mpbird 166	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A <_ 0 )
13		simp2 983	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 <_ A )
14	9, 10	letri3d 7903	. 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 929	1 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A = 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 963   = wceq 1332   e. wcel 1481   A.wral 2417   class class class wbr 3937   RRcr 7643   0cc0 7644   < clt 7824   <_ cle 7825

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1re 7738  ax-addrcl 7741  ax-rnegex 7753  ax-pre-ltirr 7756  ax-pre-apti 7759

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830

This theorem is referenced by: (None)