Theorem squeeze0 9224

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 8392	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	3ad2ant1 1049	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. A < A )
3		breq2 4129	. . . . . . 7 |- ( x = A -> ( 0 < x <-> 0 < A ) )
4		breq2 4129	. . . . . . 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 2927	. . . . 5 |- ( ( A e. RR /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
7	6	3adant2 1047	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
8	2, 7	mtod 673	. . 3 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. 0 < A )
9		simp1 1028	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A e. RR )
10		0red 8317	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 e. RR )
11	9, 10	lenltd 8434	. . 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 1029	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 <_ A )
14	9, 10	letri3d 8431	. 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 957	1 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A = 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4125   RRcr 8168   0cc0 8169   < clt 8350   <_ cle 8351

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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-apti 8284

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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356

This theorem is referenced by: (None)