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Theorem ico0 10962
Description: An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
Assertion
Ref Expression
ico0  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )

Proof of Theorem ico0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 icoval 10954 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) } )
21eqeq1d 2444 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/) ) )
3 df-ne 2601 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =/=  (/)  <->  -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/) )
4 rabn0 3647 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =/=  (/)  <->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B ) )
53, 4bitr3i 243 . . . . 5  |-  ( -. 
{ x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) )
6 xrlelttr 10746 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <_  x  /\  x  <  B )  ->  A  <  B
) )
763com23 1159 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <_  x  /\  x  <  B )  ->  A  <  B
) )
873expa 1153 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  <_  x  /\  x  <  B
)  ->  A  <  B ) )
98rexlimdva 2830 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  ->  A  <  B ) )
10 qbtwnxr 10786 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
11 qre 10579 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  e.  RR )
1211rexrd 9134 . . . . . . . . . . . 12  |-  ( x  e.  QQ  ->  x  e.  RR* )
1312a1i 11 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( x  e.  QQ  ->  x  e.  RR* )
)
14 simpr1 963 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  ->  A  e.  RR* )
15 simpl 444 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  ->  x  e.  RR* )
16 xrltle 10742 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <  x  ->  A  <_  x ) )
1714, 15, 16syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( A  <  x  ->  A  <_  x )
)
1817anim1d 548 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( ( A  < 
x  /\  x  <  B )  ->  ( A  <_  x  /\  x  < 
B ) ) )
1913, 18anim12d 547 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( ( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  -> 
( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) )
2019ex 424 . . . . . . . . . . . 12  |-  ( x  e.  RR*  ->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) ) )
2112, 20syl 16 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  (
( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) ) )
2221adantr 452 . . . . . . . . . 10  |-  ( ( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( ( A  e.  RR*  /\  B  e. 
RR*  /\  A  <  B )  ->  ( (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e. 
RR*  /\  ( A  <_  x  /\  x  < 
B ) ) ) ) )
2322pm2.43b 48 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) )
2423reximdv2 2815 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( E. x  e.  QQ  ( A  <  x  /\  x  <  B )  ->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B ) ) )
2510, 24mpd 15 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) )
26253expia 1155 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) ) )
279, 26impbid 184 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  <->  A  <  B ) )
285, 27syl5bb 249 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  A  <  B ) )
29 xrltnle 9144 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  B  <_  A ) )
3028, 29bitrd 245 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  -.  B  <_  A ) )
3130con4bid 285 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  B  <_  A ) )
322, 31bitrd 245 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709   (/)c0 3628   class class class wbr 4212  (class class class)co 6081   RR*cxr 9119    < clt 9120    <_ cle 9121   QQcq 10574   [,)cico 10918
This theorem is referenced by:  icombl  19458  ioombl  19459  difioo  24145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-ico 10922
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