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Theorem squeeze0 7945
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 7154 . . . . 5  |-  ( A  e.  RR  ->  -.  A  <  A )
213ad2ant1 936 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  -.  A  <  A )
3 breq2 3796 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
4 breq2 3796 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
53, 4imbi12d 227 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
65rspcva 2671 . . . . 5  |-  ( ( A  e.  RR  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( 0  < 
A  ->  A  <  A ) )
763adant2 934 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( 0  < 
A  ->  A  <  A ) )
82, 7mtod 599 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  -.  0  <  A )
9 simp1 915 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  e.  RR )
10 0red 7086 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  0  e.  RR )
119, 10lenltd 7193 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( A  <_ 
0  <->  -.  0  <  A ) )
128, 11mpbird 160 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  <_  0
)
13 simp2 916 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  0  <_  A
)
149, 10letri3d 7192 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( A  =  0  <->  ( A  <_ 
0  /\  0  <_  A ) ) )
1512, 13, 14mpbir2and 862 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 896    = wceq 1259    e. wcel 1409   A.wral 2323   class class class wbr 3792   RRcr 6946   0cc0 6947    < clt 7119    <_ cle 7120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1re 7036  ax-addrcl 7039  ax-rnegex 7051  ax-pre-ltirr 7054  ax-pre-apti 7057
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125
This theorem is referenced by: (None)
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