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Theorem ioo0 10834
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iooval 10833 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) } )
21eqeq1d 2374 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/) ) )
3 df-ne 2531 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =/=  (/)  <->  -.  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/) )
4 rabn0 3562 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =/=  (/)  <->  E. x  e.  RR*  ( A  <  x  /\  x  <  B ) )
53, 4bitr3i 242 . . . . 5  |-  ( -. 
{ x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/)  <->  E. x  e.  RR*  ( A  < 
x  /\  x  <  B ) )
6 xrlttr 10626 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <  x  /\  x  <  B )  ->  A  <  B
) )
763com23 1158 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <  x  /\  x  <  B )  ->  A  <  B
) )
873expa 1152 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  < 
x  /\  x  <  B )  ->  A  <  B ) )
98rexlimdva 2752 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <  x  /\  x  <  B )  ->  A  <  B ) )
10 qbtwnxr 10679 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
11 qre 10472 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR )
1211rexrd 9028 . . . . . . . . . 10  |-  ( x  e.  QQ  ->  x  e.  RR* )
1312anim1i 551 . . . . . . . . 9  |-  ( ( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e. 
RR*  /\  ( A  <  x  /\  x  < 
B ) ) )
1413reximi2 2734 . . . . . . . 8  |-  ( E. x  e.  QQ  ( A  <  x  /\  x  <  B )  ->  E. x  e.  RR*  ( A  < 
x  /\  x  <  B ) )
1510, 14syl 15 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  RR*  ( A  < 
x  /\  x  <  B ) )
16153expia 1154 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  RR*  ( A  < 
x  /\  x  <  B ) ) )
179, 16impbid 183 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <  x  /\  x  <  B )  <->  A  <  B ) )
185, 17syl5bb 248 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/)  <->  A  <  B ) )
19 xrltnle 9038 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  B  <_  A ) )
2018, 19bitrd 244 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/)  <->  -.  B  <_  A ) )
2120con4bid 284 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <  x  /\  x  <  B ) }  =  (/)  <->  B  <_  A ) )
222, 21bitrd 244 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A (,) B
)  =  (/)  <->  B  <_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   {crab 2632   (/)c0 3543   class class class wbr 4125  (class class class)co 5981   RR*cxr 9013    < clt 9014    <_ cle 9015   QQcq 10467   (,)cioo 10809
This theorem is referenced by:  ioon0  10835  iooid  10837  bndth  18671  ioombl  19137  itgsubstlem  19610  ioovolcl  27333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-q 10468  df-ioo 10813
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