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Theorem iccid 11681
Description: A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
Assertion
Ref Expression
iccid  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )

Proof of Theorem iccid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elicc1 11680 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR* )  ->  (
x  e.  ( A [,] A )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A
) ) )
21anidms 649 . . 3  |-  ( A  e.  RR*  ->  ( x  e.  ( A [,] A )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A
) ) )
3 xrlenlt 9698 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  <->  -.  x  <  A ) )
4 xrlenlt 9698 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  <_  A  <->  -.  A  <  x ) )
54ancoms 454 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
x  <_  A  <->  -.  A  <  x ) )
6 xrlttri3 11442 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  =  A  <->  ( -.  x  <  A  /\  -.  A  <  x ) ) )
76biimprd 226 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
( -.  x  < 
A  /\  -.  A  <  x )  ->  x  =  A ) )
87ancoms 454 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
( -.  x  < 
A  /\  -.  A  <  x )  ->  x  =  A ) )
98expcomd 439 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( -.  A  <  x  -> 
( -.  x  < 
A  ->  x  =  A ) ) )
105, 9sylbid 218 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
x  <_  A  ->  ( -.  x  <  A  ->  x  =  A ) ) )
1110com23 81 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( -.  x  <  A  -> 
( x  <_  A  ->  x  =  A ) ) )
123, 11sylbid 218 . . . . . . 7  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  ->  (
x  <_  A  ->  x  =  A ) ) )
1312ex 435 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  e.  RR*  ->  ( A  <_  x  ->  (
x  <_  A  ->  x  =  A ) ) ) )
14133impd 1219 . . . . 5  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  ->  x  =  A ) )
15 eleq1a 2512 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  e.  RR* ) )
16 xrleid 11449 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
17 breq2 4430 . . . . . . 7  |-  ( x  =  A  ->  ( A  <_  x  <->  A  <_  A ) )
1816, 17syl5ibrcom 225 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  A  <_  x ) )
19 breq1 4429 . . . . . . 7  |-  ( x  =  A  ->  (
x  <_  A  <->  A  <_  A ) )
2016, 19syl5ibrcom 225 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  <_  A ) )
2115, 18, 203jcad 1186 . . . . 5  |-  ( A  e.  RR*  ->  ( x  =  A  ->  (
x  e.  RR*  /\  A  <_  x  /\  x  <_  A ) ) )
2214, 21impbid 193 . . . 4  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  <->  x  =  A ) )
23 elsn 4016 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2422, 23syl6bbr 266 . . 3  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  <->  x  e.  { A } ) )
252, 24bitrd 256 . 2  |-  ( A  e.  RR*  ->  ( x  e.  ( A [,] A )  <->  x  e.  { A } ) )
2625eqrdv 2426 1  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {csn 4002   class class class wbr 4426  (class class class)co 6305   RR*cxr 9673    < clt 9674    <_ cle 9675   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-icc 11642
This theorem is referenced by:  snunioo  11756  snunico  11757  snunioc  11758  prunioo  11759  icccmplem1  21751  ivthicc  22290  ioombl  22395  volivth  22442  mbfimasn  22467  itgspliticc  22671  dvivth  22839  cvmliftlem10  29805  mblfinlem2  31682  areacirc  31741  ioounsn  35793  iocinico  35795  iocmbl  35796  snunioo2  37191  snunioo1  37198  cncfiooicc  37344
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