Theorem squeeze0 8799

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 7975	. . . . 5 |- ( A e. RR -> -. A < A )
2	1	3ad2ant1 1008	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> -. A < A )
3		breq2 3986	. . . . . . 7 |- ( x = A -> ( 0 < x <-> 0 < A ) )
4		breq2 3986	. . . . . . 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 2828	. . . . 5 |- ( ( A e. RR /\ A. x e. RR ( 0 < x -> A < x ) ) -> ( 0 < A -> A < A ) )
7	6	3adant2 1006	. . . 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 987	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A e. RR )
10		0red 7900	. . . 4 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 e. RR )
11	9, 10	lenltd 8016	. . 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 988	. 2 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> 0 <_ A )
14	9, 10	letri3d 8014	. 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 934	1 |- ( ( A e. RR /\ 0 <_ A /\ A. x e. RR ( 0 < x -> A < x ) ) -> A = 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   class class class wbr 3982   RRcr 7752   0cc0 7753   < clt 7933   <_ cle 7934

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850  ax-rnegex 7862  ax-pre-ltirr 7865  ax-pre-apti 7868

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939

This theorem is referenced by: (None)