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Theorem ioo0 9683
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 9682 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
21eqeq1d 2090 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/) ) )
3 df-ne 2201 . . . . . 6  |-  ( {
x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =/=  (/)  <->  -.  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/) )
4 rabn0 3107 . . . . . 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 9483 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <  x  /\  x  <  B )  ->  A  <  B
) )
763com23 1125 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <  x  /\  x  <  B )  ->  A  <  B
) )
873expa 1119 . . . . . . 7  |-  ( (
( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  < 
x  /\  x  <  B )  ->  A  <  B ) )
98rexlimdva 2406 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <  x  /\  x  <  B )  ->  A  <  B ) )
10 qbtwnxr 9532 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
11 qre 9394 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR )
12 rexr 8263 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  RR* )
1311, 12syl 15 . . . . . . . . . 10  |-  ( x  e.  QQ  ->  x  e.  RR* )
1413anim1i 546 . . . . . . . . 9  |-  ( (
x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e. 
RR*  /\  ( A  <  x  /\  x  < 
B ) ) )
1514reximi2 2388 . . . . . . . 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 1121 . . . . . 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 8273 . . . 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 357    /\ w3a 903    = wceq 1536    e. wcel 1538    =/= wne 2199   E.wrex 2292   {crab 2294   (/)c0 3088   class class class wbr 3600  (class class class)co 5354   RRcr 8136    <_ cle 8250   RR*cxr 8253    < clt 8254   QQcq 8379   (,)cioo 9658
This theorem is referenced by:  ioon0  9684  iooid  9686  bndth  16174  ioombl  16638  itgsubstlem  17056  oisbmi  22051  oisbmj  22052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1451  ax-6 1452  ax-7 1453  ax-gen 1454  ax-8 1540  ax-11 1541  ax-13 1542  ax-14 1543  ax-17 1545  ax-12o 1578  ax-10 1592  ax-9 1598  ax-4 1606  ax-16 1793  ax-ext 2064  ax-sep 3715  ax-nul 3723  ax-pow 3759  ax-pr 3783  ax-un 4075  ax-cnex 8192  ax-resscn 8193  ax-1cn 8194  ax-icn 8195  ax-addcl 8196  ax-addrcl 8197  ax-mulcl 8198  ax-mulrcl 8199  ax-mulcom 8200  ax-addass 8201  ax-mulass 8202  ax-distr 8203  ax-i2m1 8204  ax-1ne0 8205  ax-1rid 8206  ax-rnegex 8207  ax-rrecex 8208  ax-cnre 8209  ax-pre-lttri 8210  ax-pre-lttrn 8211  ax-pre-ltadd 8212  ax-pre-mulgt0 8213  ax-pre-sup 8214
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3or 904  df-3an 905  df-tru 1268  df-ex 1456  df-sb 1754  df-eu 1976  df-mo 1977  df-clab 2070  df-cleq 2075  df-clel 2078  df-ne 2201  df-nel 2202  df-ral 2295  df-rex 2296  df-reu 2297  df-rab 2298  df-v 2494  df-sbc 2668  df-csb 2750  df-dif 2813  df-un 2815  df-in 2817  df-ss 2821  df-pss 2823  df-nul 3089  df-if 3199  df-pw 3260  df-sn 3278  df-pr 3279  df-tp 3280  df-op 3281  df-uni 3439  df-iun 3516  df-br 3601  df-opab 3655  df-mpt 3656  df-tr 3688  df-eprel 3870  df-id 3874  df-po 3879  df-so 3880  df-fr 3917  df-we 3919  df-ord 3960  df-on 3961  df-lim 3962  df-suc 3963  df-om 4243  df-xp 4289  df-rel 4290  df-cnv 4291  df-co 4292  df-dm 4293  df-rn 4294  df-res 4295  df-ima 4296  df-fun 4297  df-fn 4298  df-f 4299  df-f1 4300  df-fo 4301  df-f1o 4302  df-fv 4303  df-ov 5357  df-oprab 5358  df-mpt2 5359  df-1st 5608  df-2nd 5609  df-iota 5762  df-recs 5835  df-rdg 5870  df-er 6102  df-en 6289  df-dom 6290  df-sdom 6291  df-riota 6455  df-sup 6656  df-pnf 8255  df-mnf 8256  df-xr 8257  df-ltxr 8258  df-le 8259  df-sub 8392  df-neg 8393  df-div 8621  df-n 8863  df-n0 9056  df-z 9108  df-uz 9305  df-q 9390  df-ioo 9662
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