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Theorem ico0 11689
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 11681 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) } )
21eqeq1d 2455 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/) ) )
3 df-ne 2626 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =/=  (/)  <->  -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/) )
4 rabn0 3754 . . . . . 6  |-  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =/=  (/)  <->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B ) )
53, 4bitr3i 255 . . . . 5  |-  ( -. 
{ x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) )
6 xrlelttr 11460 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR*  /\  B  e. 
RR* )  ->  (
( A  <_  x  /\  x  <  B )  ->  A  <  B
) )
763com23 1215 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  x  e. 
RR* )  ->  (
( A  <_  x  /\  x  <  B )  ->  A  <  B
) )
873expa 1209 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  RR* )  ->  ( ( A  <_  x  /\  x  <  B
)  ->  A  <  B ) )
98rexlimdva 2881 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  ->  A  <  B ) )
10 qbtwnxr 11500 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
11 qre 11276 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  e.  RR )
1211rexrd 9695 . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  ->  A  e.  RR* )
15 simpl 459 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  ->  x  e.  RR* )
16 xrltle 11455 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <  x  ->  A  <_  x ) )
1714, 15, 16syl2anc 667 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( A  <  x  ->  A  <_  x )
)
1817anim1d 568 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B ) )  -> 
( ( A  < 
x  /\  x  <  B )  ->  ( A  <_  x  /\  x  < 
B ) ) )
1913, 18anim12d 566 . . . . . . . . . . . . 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 436 . . . . . . . . . . . 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 17 . . . . . . . . . . 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 467 . . . . . . . . . 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 52 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( x  e.  QQ  /\  ( A  <  x  /\  x  <  B ) )  ->  ( x  e.  RR*  /\  ( A  <_  x  /\  x  <  B ) ) ) )
2423reximdv2 2860 . . . . . . . 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 1211 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  RR*  ( A  <_  x  /\  x  <  B
) ) )
279, 26impbid 194 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  RR*  ( A  <_  x  /\  x  <  B )  <->  A  <  B ) )
285, 27syl5bb 261 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  A  <  B ) )
29 xrltnle 9706 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -.  B  <_  A ) )
3028, 29bitrd 257 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( -.  { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  -.  B  <_  A ) )
3130con4bid 295 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( { x  e.  RR*  |  ( A  <_  x  /\  x  <  B ) }  =  (/)  <->  B  <_  A ) )
322, 31bitrd 257 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   E.wrex 2740   {crab 2743   (/)c0 3733   class class class wbr 4405  (class class class)co 6295   RR*cxr 9679    < clt 9680    <_ cle 9681   QQcq 11271   [,)cico 11644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-ico 11648
This theorem is referenced by:  icombl  22529  ioombl  22530  difioo  28376  volico  38373  ovn0lem  38397  ovnhoilem1  38433  hspmbllem1  38458
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