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Theorem squeeze0 10468
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 9613 . . . 4  |-  0  e.  RR
2 leloe 9688 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
31, 2mpan 670 . . 3  |-  ( A  e.  RR  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
4 breq2 4460 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
5 breq2 4460 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
64, 5imbi12d 320 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
76rspcv 3206 . . . . 5  |-  ( A  e.  RR  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  ( 0  < 
A  ->  A  <  A ) ) )
8 ltnr 9696 . . . . . . . . 9  |-  ( A  e.  RR  ->  -.  A  <  A )
98pm2.21d 106 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  A  ->  A  =  0 ) )
109com12 31 . . . . . . 7  |-  ( A  <  A  ->  ( A  e.  RR  ->  A  =  0 ) )
1110imim2i 14 . . . . . 6  |-  ( ( 0  <  A  ->  A  <  A )  -> 
( 0  <  A  ->  ( A  e.  RR  ->  A  =  0 ) ) )
1211com13 80 . . . . 5  |-  ( A  e.  RR  ->  (
0  <  A  ->  ( ( 0  <  A  ->  A  <  A )  ->  A  =  0 ) ) )
137, 12syl5d 67 . . . 4  |-  ( A  e.  RR  ->  (
0  <  A  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
14 ax-1 6 . . . . . 6  |-  ( A  =  0  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) )
1514eqcoms 2469 . . . . 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 380 . . 3  |-  ( A  e.  RR  ->  (
( 0  <  A  \/  0  =  A
)  ->  ( A. x  e.  RR  (
0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
183, 17sylbid 215 . 2  |-  ( A  e.  RR  ->  (
0  <_  A  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
19183imp 1190 1  |-  ( ( A  e.  RR  /\  0  <_  A  /\  A. x  e.  RR  (
0  <  x  ->  A  <  x ) )  ->  A  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   class class class wbr 4456   RRcr 9508   0cc0 9509    < clt 9645    <_ cle 9646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651
This theorem is referenced by: (None)
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