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Theorem ioc0 9670
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 9334 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,] B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) } )
21eqeq1d 2096 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,] B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) }  =  (/) ) )
3 xrltletr 9270 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <  x  /\  x  <_  B )  ->  A  <  B
) )
433com23 1149 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <  x  /\  x  <_  B )  ->  A  <  B
) )
543expa 1143 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  < 
x  /\  x  <_  B )  ->  A  <  B ) )
65rexlimdva 2489 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <  x  /\  x  <_  B )  ->  A  <  B ) )
7 qbtwnxr 9665 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
8 qre 9108 . . . . . . . . . . . 12  |-  ( x  e.  QQ  ->  x  e.  RR )
98rexrd 7535 . . . . . . . . . . 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 9266 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  B  e.  RR* )  ->  (
x  <  B  ->  x  <_  B ) )
12113ad2antr2 1109 . . . . . . . . . . . . . 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 2472 . . . . . . 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 1145 . . . . 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 627 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  E. x  e.  RR*  ( A  <  x  /\  x  <_  B )  <->  -.  A  <  B ) )
24 rabeq0 3312 . . . . 5  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <_  B ) }  =  (/)  <->  A. x  e.  RR*  -.  ( A  <  x  /\  x  <_  B ) )
25 ralnex 2369 . . . . 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 7549 . . . 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 924    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   (/)c0 3286   class class class wbr 3845  (class class class)co 5652   RR*cxr 7519    < clt 7520    <_ cle 7521   QQcq 9102   (,]cioc 9305
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-po 4123  df-iso 4124  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-ioc 9309
This theorem is referenced by: (None)
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