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Theorem squeeze0 8813
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 7989 . . . . 5  |-  ( A  e.  RR  ->  -.  A  <  A )
213ad2ant1 1013 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  -.  A  <  A )
3 breq2 3991 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
4 breq2 3991 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
53, 4imbi12d 233 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
65rspcva 2832 . . . . 5  |-  ( ( A  e.  RR  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( 0  < 
A  ->  A  <  A ) )
763adant2 1011 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( 0  < 
A  ->  A  <  A ) )
82, 7mtod 658 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  -.  0  <  A )
9 simp1 992 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  e.  RR )
10 0red 7914 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  0  e.  RR )
119, 10lenltd 8030 . . 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 993 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  0  <_  A
)
149, 10letri3d 8028 . 2  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  ( A  =  0  <->  ( A  <_ 
0  /\  0  <_  A ) ) )
1512, 13, 14mpbir2and 939 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 973    = wceq 1348    e. wcel 2141   A.wral 2448   class class class wbr 3987   RRcr 7766   0cc0 7767    < clt 7947    <_ cle 7948
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1re 7861  ax-addrcl 7864  ax-rnegex 7876  ax-pre-ltirr 7879  ax-pre-apti 7882
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-cnv 4617  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953
This theorem is referenced by: (None)
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