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Theorem ioo0 9702
Description: An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
Assertion
Ref Expression
ioo0  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )

Proof of Theorem ioo0
StepHypRef Expression
1 iooval 9701 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
21eqeq1d 2069 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/) ) )
3 df-ne 2181 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =/=  (/)  <->  -.  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/) )
4 rabn0 3088 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =/=  (/)  <->  E. x  e.  RR*  ( A  <  x  /\  x  <  B ) )
53, 4bitr3i 240 . . . . 5  |-  ( -. 
{ x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/)  <->  E. x  e.  RR*  ( A  < 
x  /\  x  <  B ) )
6 xrlttr 9499 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <  x  /\  x  <  B )  ->  A  <  B
) )
763com23 1113 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <  x  /\  x  <  B )  ->  A  <  B
) )
873expa 1107 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  < 
x  /\  x  <  B )  ->  A  <  B ) )
98rexlimdva 2386 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <  x  /\  x  <  B )  ->  A  <  B ) )
10 qbtwnxr 9551 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
11 qre 9410 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR )
12 rexr 8277 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  RR* )
1311, 12syl 15 . . . . . . . . . 10  |-  ( x  e.  QQ  ->  x  e.  RR* )
1413anim1i 543 . . . . . . . . 9  |-  ( ( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e. 
RR*  /\  ( A  <  x  /\  x  < 
B ) ) )
1514reximi2 2368 . . . . . . . 8  |-  ( E. x  e.  QQ  ( A  <  x  /\  x  <  B )  ->  E. x  e.  RR*  ( A  < 
x  /\  x  <  B ) )
1610, 15syl 15 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  RR*  ( A  < 
x  /\  x  <  B ) )
17163expia 1109 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  RR*  ( A  < 
x  /\  x  <  B ) ) )
189, 17impbid 181 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <  x  /\  x  <  B )  <->  A  <  B ) )
195, 18syl5bb 246 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/)  <->  A  <  B ) )
20 xrltnle 8287 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  B  <_  A ) )
2119, 20bitrd 242 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/)  <->  -.  B  <_  A ) )
2221con4bid 282 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/)  <->  B  <_  A ) )
232, 22bitrd 242 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 174    /\ wa 356    /\ w3a 893    = wceq 1517    e. wcel 1519    =/= wne 2179   E.wrex 2272   {crab 2274   (/)c0 3069   class class class wbr 3583  (class class class)co 5355   RRcr 8150    <_ cle 8264   RR*cxr 8267    < clt 8268   QQcq 8393   (,)cioo 9677
This theorem is referenced by:  ioon0  9703  iooid  9705  bndth  16530  ioombl  16994  itgsubstlem  17412  oisbmi  22393  oisbmj  22394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1439  ax-6 1440  ax-7 1441  ax-gen 1442  ax-8 1521  ax-11 1522  ax-13 1523  ax-14 1524  ax-17 1526  ax-12o 1559  ax-10 1573  ax-9 1579  ax-4 1586  ax-16 1772  ax-ext 2043  ax-sep 3698  ax-nul 3706  ax-pow 3742  ax-pr 3766  ax-un 4058  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-icn 8209  ax-addcl 8210  ax-addrcl 8211  ax-mulcl 8212  ax-mulrcl 8213  ax-mulcom 8214  ax-addass 8215  ax-mulass 8216  ax-distr 8217  ax-i2m1 8218  ax-1ne0 8219  ax-1rid 8220  ax-rnegex 8221  ax-rrecex 8222  ax-cnre 8223  ax-pre-lttri 8224  ax-pre-lttrn 8225  ax-pre-ltadd 8226  ax-pre-mulgt0 8227  ax-pre-sup 8228
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 894  df-3an 895  df-tru 1256  df-ex 1444  df-sb 1733  df-eu 1955  df-mo 1956  df-clab 2049  df-cleq 2054  df-clel 2057  df-ne 2181  df-nel 2182  df-ral 2275  df-rex 2276  df-reu 2277  df-rab 2278  df-v 2474  df-sbc 2648  df-csb 2730  df-dif 2793  df-un 2795  df-in 2797  df-ss 2801  df-pss 2803  df-nul 3070  df-if 3179  df-pw 3240  df-sn 3258  df-pr 3259  df-tp 3260  df-op 3261  df-uni 3422  df-iun 3499  df-br 3584  df-opab 3638  df-mpt 3639  df-tr 3671  df-eprel 3853  df-id 3857  df-po 3862  df-so 3863  df-fr 3900  df-we 3902  df-ord 3943  df-on 3944  df-lim 3945  df-suc 3946  df-om 4221  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-rn 4272  df-res 4273  df-ima 4274  df-fun 4275  df-fn 4276  df-f 4277  df-f1 4278  df-fo 4279  df-f1o 4280  df-fv 4281  df-ov 5358  df-oprab 5359  df-mpt2 5360  df-1st 5609  df-2nd 5610  df-iota 5764  df-recs 5837  df-rdg 5872  df-er 6109  df-en 6296  df-dom 6297  df-sdom 6298  df-riota 6462  df-sup 6670  df-pnf 8269  df-mnf 8270  df-xr 8271  df-ltxr 8272  df-le 8273  df-sub 8406  df-neg 8407  df-div 8637  df-n 8879  df-n0 9072  df-z 9124  df-uz 9321  df-q 9406  df-ioo 9681
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