MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  squeeze0 Structured version   Unicode version

Theorem squeeze0 9918
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 9096 . . . 4  |-  0  e.  RR
2 leloe 9166 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
31, 2mpan 653 . . 3  |-  ( A  e.  RR  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
4 breq2 4219 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
5 breq2 4219 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
64, 5imbi12d 313 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
76rspcv 3050 . . . . 5  |-  ( A  e.  RR  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  ( 0  < 
A  ->  A  <  A ) ) )
8 ltnr 9173 . . . . . . . . 9  |-  ( A  e.  RR  ->  -.  A  <  A )
98pm2.21d 101 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  A  ->  A  =  0 ) )
109com12 30 . . . . . . 7  |-  ( A  <  A  ->  ( A  e.  RR  ->  A  =  0 ) )
1110imim2i 14 . . . . . 6  |-  ( ( 0  <  A  ->  A  <  A )  -> 
( 0  <  A  ->  ( A  e.  RR  ->  A  =  0 ) ) )
1211com13 77 . . . . 5  |-  ( A  e.  RR  ->  (
0  <  A  ->  ( ( 0  <  A  ->  A  <  A )  ->  A  =  0 ) ) )
137, 12syl5d 65 . . . 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 2441 . . . . 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 371 . . 3  |-  ( A  e.  RR  ->  (
( 0  <  A  \/  0  =  A
)  ->  ( A. x  e.  RR  (
0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
183, 17sylbid 208 . 2  |-  ( A  e.  RR  ->  (
0  <_  A  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  A  =  0 ) ) )
19183imp 1148 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 178    \/ wo 359    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4215   RRcr 8994   0cc0 8995    < clt 9125    <_ cle 9126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-i2m1 9063  ax-1ne0 9064  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131
  Copyright terms: Public domain W3C validator