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Theorem squeeze0 8874
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
StepHypRef Expression
1 ltnr 8047 . . . . 5  |-  ( A  e.  RR  ->  -.  A  <  A )
213ad2ant1 1019 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  -.  A  <  A )
3 breq2 4019 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
4 breq2 4019 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
53, 4imbi12d 234 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
65rspcva 2851 . . . . 5  |-  ( ( A  e.  RR  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( 0  < 
A  ->  A  <  A ) )
763adant2 1017 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( 0  < 
A  ->  A  <  A ) )
82, 7mtod 664 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  -.  0  <  A )
9 simp1 998 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  e.  RR )
10 0red 7971 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  0  e.  RR )
119, 10lenltd 8088 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( A  <_ 
0  <->  -.  0  <  A ) )
128, 11mpbird 167 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  <_  0
)
13 simp2 999 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  0  <_  A
)
149, 10letri3d 8086 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( A  =  0  <->  ( A  <_ 
0  /\  0  <_  A ) ) )
1512, 13, 14mpbir2and 945 1  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 979    = wceq 1363    e. wcel 2158   A.wral 2465   class class class wbr 4015   RRcr 7823   0cc0 7824    < clt 8005    <_ cle 8006
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-apti 7939
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011
This theorem is referenced by: (None)
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