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Theorem ico0 11334
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 11326 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) } )
21eqeq1d 2441 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/) ) )
3 df-ne 2598 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =/=  (/)  <->  -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/) )
4 rabn0 3645 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =/=  (/)  <->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B ) )
53, 4bitr3i 251 . . . . 5  |-  ( -. 
{ x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) )
6 xrlelttr 11118 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <_  x  /\  x  <  B )  ->  A  <  B
) )
763com23 1186 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <_  x  /\  x  <  B )  ->  A  <  B
) )
873expa 1180 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  <_  x  /\  x  <  B
)  ->  A  <  B ) )
98rexlimdva 2831 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  ->  A  <  B ) )
10 qbtwnxr 11158 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
11 qre 10946 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  e.  RR )
1211rexrd 9421 . . . . . . . . . . . 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 987 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  ->  A  e.  RR* )
15 simpl 454 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  ->  x  e.  RR* )
16 xrltle 11114 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <  x  ->  A  <_  x ) )
1714, 15, 16syl2anc 654 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( A  <  x  ->  A  <_  x )
)
1817anim1d 559 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( ( A  < 
x  /\  x  <  B )  ->  ( A  <_  x  /\  x  < 
B ) ) )
1913, 18anim12d 558 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 462 . . . . . . . . . 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 50 . . . . . . . . 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 1182 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) ) )
279, 26impbid 191 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  <->  A  <  B ) )
285, 27syl5bb 257 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  A  <  B ) )
29 xrltnle 9431 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  B  <_  A ) )
3028, 29bitrd 253 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  -.  B  <_  A ) )
3130con4bid 293 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  B  <_  A ) )
322, 31bitrd 253 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706   {crab 2709   (/)c0 3625   class class class wbr 4280  (class class class)co 6080   RR*cxr 9405    < clt 9406    <_ cle 9407   QQcq 10941   [,)cico 11290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-ico 11294
This theorem is referenced by:  icombl  20887  ioombl  20888  difioo  25895
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