ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  squeeze0 Unicode version

Theorem squeeze0 8669
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 7848 . . . . 5  |-  ( A  e.  RR  ->  -.  A  <  A )
213ad2ant1 1002 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  -.  A  <  A )
3 breq2 3933 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
4 breq2 3933 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
53, 4imbi12d 233 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
65rspcva 2787 . . . . 5  |-  ( ( A  e.  RR  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( 0  < 
A  ->  A  <  A ) )
763adant2 1000 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( 0  < 
A  ->  A  <  A ) )
82, 7mtod 652 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  -.  0  <  A )
9 simp1 981 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  e.  RR )
10 0red 7774 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  0  e.  RR )
119, 10lenltd 7887 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( A  <_ 
0  <->  -.  0  <  A ) )
128, 11mpbird 166 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  <_  0
)
13 simp2 982 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  0  <_  A
)
149, 10letri3d 7886 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( A  =  0  <->  ( A  <_ 
0  /\  0  <_  A ) ) )
1512, 13, 14mpbir2and 928 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 962    = wceq 1331    e. wcel 1480   A.wral 2416   class class class wbr 3929   RRcr 7626   0cc0 7627    < clt 7807    <_ cle 7808
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1re 7721  ax-addrcl 7724  ax-rnegex 7736  ax-pre-ltirr 7739  ax-pre-apti 7742
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator