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Theorem iccid 10701
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 10700 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR* )  ->  (
x  e.  ( A [,] A )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A
) ) )
21anidms 626 . . 3  |-  ( A  e.  RR*  ->  ( x  e.  ( A [,] A )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A
) ) )
3 xrlenlt 8890 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  <->  -.  x  <  A ) )
4 xrlenlt 8890 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  <_  A  <->  -.  A  <  x ) )
54ancoms 439 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
x  <_  A  <->  -.  A  <  x ) )
6 xrlttri3 10477 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  =  A  <->  ( -.  x  <  A  /\  -.  A  <  x ) ) )
76biimprd 214 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
( -.  x  < 
A  /\  -.  A  <  x )  ->  x  =  A ) )
87ancoms 439 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
( -.  x  < 
A  /\  -.  A  <  x )  ->  x  =  A ) )
98exp3acom23 1362 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( -.  A  <  x  -> 
( -.  x  < 
A  ->  x  =  A ) ) )
105, 9sylbid 206 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
x  <_  A  ->  ( -.  x  <  A  ->  x  =  A ) ) )
1110com23 72 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( -.  x  <  A  -> 
( x  <_  A  ->  x  =  A ) ) )
123, 11sylbid 206 . . . . . . 7  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  ->  (
x  <_  A  ->  x  =  A ) ) )
1312ex 423 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  e.  RR*  ->  ( A  <_  x  ->  (
x  <_  A  ->  x  =  A ) ) ) )
14133impd 1165 . . . . 5  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  ->  x  =  A ) )
15 eleq1a 2352 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  e.  RR* ) )
16 xrleid 10484 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
17 breq2 4027 . . . . . . 7  |-  ( x  =  A  ->  ( A  <_  x  <->  A  <_  A ) )
1816, 17syl5ibrcom 213 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  A  <_  x ) )
19 breq1 4026 . . . . . . 7  |-  ( x  =  A  ->  (
x  <_  A  <->  A  <_  A ) )
2016, 19syl5ibrcom 213 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  <_  A ) )
2115, 18, 203jcad 1133 . . . . 5  |-  ( A  e.  RR*  ->  ( x  =  A  ->  (
x  e.  RR*  /\  A  <_  x  /\  x  <_  A ) ) )
2214, 21impbid 183 . . . 4  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  <->  x  =  A ) )
23 elsn 3655 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2422, 23syl6bbr 254 . . 3  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  <->  x  e.  { A } ) )
252, 24bitrd 244 . 2  |-  ( A  e.  RR*  ->  ( x  e.  ( A [,] A )  <->  x  e.  { A } ) )
2625eqrdv 2281 1  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {csn 3640   class class class wbr 4023  (class class class)co 5858   RR*cxr 8866    < clt 8867    <_ cle 8868   [,]cicc 10659
This theorem is referenced by:  snunioo  10762  snunico  10763  prunioo  10764  icccmplem1  18327  ivthicc  18818  ioombl  18922  volivth  18962  mbfimasn  18989  itgspliticc  19191  dvivth  19357  snunioc  23267  cvmliftlem10  23825  areacirc  24931  ccidOLD  26229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
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