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Theorem squeeze0 9877
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 0re 9055 . . . 4  |-  0  e.  RR
2 leloe 9125 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
31, 2mpan 652 . . 3  |-  ( A  e.  RR  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
4 breq2 4184 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
5 breq2 4184 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
64, 5imbi12d 312 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
76rspcv 3016 . . . . 5  |-  ( A  e.  RR  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  ( 0  < 
A  ->  A  <  A ) ) )
8 ltnr 9132 . . . . . . . . 9  |-  ( A  e.  RR  ->  -.  A  <  A )
98pm2.21d 100 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  A  ->  A  =  0 ) )
109com12 29 . . . . . . 7  |-  ( A  <  A  ->  ( A  e.  RR  ->  A  =  0 ) )
1110imim2i 14 . . . . . 6  |-  ( ( 0  <  A  ->  A  <  A )  -> 
( 0  <  A  ->  ( A  e.  RR  ->  A  =  0 ) ) )
1211com13 76 . . . . 5  |-  ( A  e.  RR  ->  (
0  <  A  ->  ( ( 0  <  A  ->  A  <  A )  ->  A  =  0 ) ) )
137, 12syl5d 64 . . . 4  |-  ( A  e.  RR  ->  (
0  <  A  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
14 ax-1 5 . . . . . 6  |-  ( A  =  0  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) )
1514eqcoms 2415 . . . . 5  |-  ( 0  =  A  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) )
1615a1i 11 . . . 4  |-  ( A  e.  RR  ->  (
0  =  A  -> 
( A. x  e.  RR  ( 0  < 
x  ->  A  <  x )  ->  A  = 
0 ) ) )
1713, 16jaod 370 . . 3  |-  ( A  e.  RR  ->  (
( 0  <  A  \/  0  =  A
)  ->  ( A. x  e.  RR  (
0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
183, 17sylbid 207 . 2  |-  ( A  e.  RR  ->  (
0  <_  A  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
19183imp 1147 1  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674   class class class wbr 4180   RRcr 8953   0cc0 8954    < clt 9084    <_ cle 9085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-i2m1 9022  ax-1ne0 9023  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090
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