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Theorem iccid 11451
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 11450 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR* )  ->  (
x  e.  ( A [,] A )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A
) ) )
21anidms 645 . . 3  |-  ( A  e.  RR*  ->  ( x  e.  ( A [,] A )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A
) ) )
3 xrlenlt 9548 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  <->  -.  x  <  A ) )
4 xrlenlt 9548 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  <_  A  <->  -.  A  <  x ) )
54ancoms 453 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
x  <_  A  <->  -.  A  <  x ) )
6 xrlttri3 11226 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  =  A  <->  ( -.  x  <  A  /\  -.  A  <  x ) ) )
76biimprd 223 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
( -.  x  < 
A  /\  -.  A  <  x )  ->  x  =  A ) )
87ancoms 453 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
( -.  x  < 
A  /\  -.  A  <  x )  ->  x  =  A ) )
98expcomd 438 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( -.  A  <  x  -> 
( -.  x  < 
A  ->  x  =  A ) ) )
105, 9sylbid 215 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
x  <_  A  ->  ( -.  x  <  A  ->  x  =  A ) ) )
1110com23 78 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( -.  x  <  A  -> 
( x  <_  A  ->  x  =  A ) ) )
123, 11sylbid 215 . . . . . . 7  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  ->  (
x  <_  A  ->  x  =  A ) ) )
1312ex 434 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  e.  RR*  ->  ( A  <_  x  ->  (
x  <_  A  ->  x  =  A ) ) ) )
14133impd 1202 . . . . 5  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  ->  x  =  A ) )
15 eleq1a 2535 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  e.  RR* ) )
16 xrleid 11233 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
17 breq2 4399 . . . . . . 7  |-  ( x  =  A  ->  ( A  <_  x  <->  A  <_  A ) )
1816, 17syl5ibrcom 222 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  A  <_  x ) )
19 breq1 4398 . . . . . . 7  |-  ( x  =  A  ->  (
x  <_  A  <->  A  <_  A ) )
2016, 19syl5ibrcom 222 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  <_  A ) )
2115, 18, 203jcad 1169 . . . . 5  |-  ( A  e.  RR*  ->  ( x  =  A  ->  (
x  e.  RR*  /\  A  <_  x  /\  x  <_  A ) ) )
2214, 21impbid 191 . . . 4  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  <->  x  =  A ) )
23 elsn 3994 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2422, 23syl6bbr 263 . . 3  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  <->  x  e.  { A } ) )
252, 24bitrd 253 . 2  |-  ( A  e.  RR*  ->  ( x  e.  ( A [,] A )  <->  x  e.  { A } ) )
2625eqrdv 2449 1  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {csn 3980   class class class wbr 4395  (class class class)co 6195   RR*cxr 9523    < clt 9524    <_ cle 9525   [,]cicc 11409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-pre-lttri 9462  ax-pre-lttrn 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-icc 11413
This theorem is referenced by:  snunioo  11523  snunico  11524  snunioc  11525  prunioo  11526  icccmplem1  20526  ivthicc  21069  ioombl  21174  volivth  21215  mbfimasn  21240  itgspliticc  21442  dvivth  21610  cvmliftlem10  27322  mblfinlem2  28572  areacirc  28632  ioounsn  29728  iocinico  29730  iocmbl  29731
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