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Theorem squeeze0 9749
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 8928 . . . 4  |-  0  e.  RR
2 leloe 8998 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
31, 2mpan 651 . . 3  |-  ( A  e.  RR  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
4 breq2 4108 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
5 breq2 4108 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
64, 5imbi12d 311 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
76rspcv 2956 . . . . 5  |-  ( A  e.  RR  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  ( 0  < 
A  ->  A  <  A ) ) )
8 ltnr 9005 . . . . . . . . 9  |-  ( A  e.  RR  ->  -.  A  <  A )
98pm2.21d 98 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  A  ->  A  =  0 ) )
109com12 27 . . . . . . 7  |-  ( A  <  A  ->  ( A  e.  RR  ->  A  =  0 ) )
1110imim2i 13 . . . . . 6  |-  ( ( 0  <  A  ->  A  <  A )  -> 
( 0  <  A  ->  ( A  e.  RR  ->  A  =  0 ) ) )
1211com13 74 . . . . 5  |-  ( A  e.  RR  ->  (
0  <  A  ->  ( ( 0  <  A  ->  A  <  A )  ->  A  =  0 ) ) )
137, 12syl5d 62 . . . 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 2361 . . . . 5  |-  ( 0  =  A  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) )
1615a1i 10 . . . 4  |-  ( A  e.  RR  ->  (
0  =  A  -> 
( A. x  e.  RR  ( 0  < 
x  ->  A  <  x )  ->  A  = 
0 ) ) )
1713, 16jaod 369 . . 3  |-  ( A  e.  RR  ->  (
( 0  <  A  \/  0  =  A
)  ->  ( A. x  e.  RR  (
0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
183, 17sylbid 206 . 2  |-  ( A  e.  RR  ->  (
0  <_  A  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
19183imp 1145 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 176    \/ wo 357    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   class class class wbr 4104   RRcr 8826   0cc0 8827    < clt 8957    <_ cle 8958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-i2m1 8895  ax-1ne0 8896  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963
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