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Theorem ioc0 9401
Description: An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
Assertion
Ref Expression
ioc0  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,] B
)  =  (/)  <->  B  <_  A ) )

Proof of Theorem ioc0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iocval 9069 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,] B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) } )
21eqeq1d 2091 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,] B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) }  =  (/) ) )
3 xrltletr 9005 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <  x  /\  x  <_  B )  ->  A  <  B
) )
433com23 1145 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <  x  /\  x  <_  B )  ->  A  <  B
) )
543expa 1139 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  < 
x  /\  x  <_  B )  ->  A  <  B ) )
65rexlimdva 2482 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <  x  /\  x  <_  B )  ->  A  <  B ) )
7 qbtwnxr 9396 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
8 qre 8843 . . . . . . . . . . . 12  |-  ( x  e.  QQ  ->  x  e.  RR )
98rexrd 7282 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR* )
109a1i 9 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( x  e.  QQ  ->  x  e.  RR* )
)
11 xrltle 9001 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  B  e.  RR* )  ->  (
x  <  B  ->  x  <_  B ) )
12113ad2antr2 1105 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( x  <  B  ->  x  <_  B )
)
1312anim2d 330 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( ( A  < 
x  /\  x  <  B )  ->  ( A  <  x  /\  x  <_  B ) ) )
1410, 13anim12d 328 . . . . . . . . . . . 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 ) ) ) )
1514ex 113 . . . . . . . . . . 11  |-  ( 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 ) ) ) ) )
169, 15syl 14 . . . . . . . . . 10  |-  ( 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 ) ) ) ) )
1716adantr 270 . . . . . . . . 9  |-  ( ( 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 ) ) ) ) )
1817pm2.43b 51 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e.  RR*  /\  ( A  <  x  /\  x  <_  B ) ) ) )
1918reximdv2 2465 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( E. x  e.  QQ  ( A  <  x  /\  x  <  B )  ->  E. x  e.  RR*  ( A  <  x  /\  x  <_  B ) ) )
207, 19mpd 13 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  RR*  ( A  < 
x  /\  x  <_  B ) )
21203expia 1141 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  RR*  ( A  < 
x  /\  x  <_  B ) ) )
226, 21impbid 127 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <  x  /\  x  <_  B )  <->  A  <  B ) )
2322notbid 625 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  E. x  e.  RR*  ( A  <  x  /\  x  <_  B )  <->  -.  A  <  B ) )
24 rabeq0 3290 . . . . 5  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) }  =  (/)  <->  A. x  e.  RR*  -.  ( A  <  x  /\  x  <_  B ) )
25 ralnex 2363 . . . . 5  |-  ( A. x  e.  RR*  -.  ( A  <  x  /\  x  <_  B )  <->  -.  E. x  e.  RR*  ( A  < 
x  /\  x  <_  B ) )
2624, 25bitri 182 . . . 4  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) }  =  (/)  <->  -.  E. x  e.  RR*  ( A  < 
x  /\  x  <_  B ) )
2726a1i 9 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) }  =  (/)  <->  -.  E. x  e.  RR*  ( A  < 
x  /\  x  <_  B ) ) )
28 xrlenlt 7296 . . . 4  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
2928ancoms 264 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
3023, 27, 293bitr4d 218 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) }  =  (/)  <->  B  <_  A ) )
312, 30bitrd 186 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,] B
)  =  (/)  <->  B  <_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   {crab 2357   (/)c0 3267   class class class wbr 3805  (class class class)co 5563   RR*cxr 7266    < clt 7267    <_ cle 7268   QQcq 8837   (,]cioc 9040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-ioc 9044
This theorem is referenced by: (None)
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