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

Theorem squeeze0 10234
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 9385 . . . 4  |-  0  e.  RR
2 leloe 9460 . . . 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 4295 . . . . . . 7  |-  ( x  =  A  ->  (
0  <  x  <->  0  <  A ) )
5 breq2 4295 . . . . . . 7  |-  ( x  =  A  ->  ( A  <  x  <->  A  <  A ) )
64, 5imbi12d 320 . . . . . 6  |-  ( x  =  A  ->  (
( 0  <  x  ->  A  <  x )  <-> 
( 0  <  A  ->  A  <  A ) ) )
76rspcv 3068 . . . . 5  |-  ( A  e.  RR  ->  ( A. x  e.  RR  ( 0  <  x  ->  A  <  x )  ->  ( 0  < 
A  ->  A  <  A ) ) )
8 ltnr 9468 . . . . . . . . 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 2445 . . . . 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 1181 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 965    = wceq 1369    e. wcel 1756   A.wral 2714   class class class wbr 4291   RRcr 9280   0cc0 9281    < clt 9417    <_ cle 9418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-i2m1 9349  ax-1ne0 9350  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator