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Theorem iccid 8895
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 8894 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR* )  ->  (
x  e.  ( A [,] A )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A
) ) )
21anidms 383 . . 3  |-  ( A  e.  RR*  ->  ( x  e.  ( A [,] A )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A
) ) )
3 xrlenlt 7143 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  <->  -.  x  <  A ) )
4 xrlenlt 7143 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  <_  A  <->  -.  A  <  x ) )
54ancoms 259 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
x  <_  A  <->  -.  A  <  x ) )
6 xrlttri3 8819 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  =  A  <->  ( -.  x  <  A  /\  -.  A  <  x ) ) )
76biimprd 151 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
( -.  x  < 
A  /\  -.  A  <  x )  ->  x  =  A ) )
87ancoms 259 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
( -.  x  < 
A  /\  -.  A  <  x )  ->  x  =  A ) )
98expcomd 1346 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( -.  A  <  x  -> 
( -.  x  < 
A  ->  x  =  A ) ) )
105, 9sylbid 143 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  (
x  <_  A  ->  ( -.  x  <  A  ->  x  =  A ) ) )
1110com23 76 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( -.  x  <  A  -> 
( x  <_  A  ->  x  =  A ) ) )
123, 11sylbid 143 . . . . . . 7  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  ->  (
x  <_  A  ->  x  =  A ) ) )
1312ex 112 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  e.  RR*  ->  ( A  <_  x  ->  (
x  <_  A  ->  x  =  A ) ) ) )
14133impd 1129 . . . . 5  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  ->  x  =  A ) )
15 eleq1a 2125 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  e.  RR* ) )
16 xrleid 8821 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
17 breq2 3796 . . . . . . 7  |-  ( x  =  A  ->  ( A  <_  x  <->  A  <_  A ) )
1816, 17syl5ibrcom 150 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  A  <_  x ) )
19 breq1 3795 . . . . . . 7  |-  ( x  =  A  ->  (
x  <_  A  <->  A  <_  A ) )
2016, 19syl5ibrcom 150 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  <_  A ) )
2115, 18, 203jcad 1096 . . . . 5  |-  ( A  e.  RR*  ->  ( x  =  A  ->  (
x  e.  RR*  /\  A  <_  x  /\  x  <_  A ) ) )
2214, 21impbid 124 . . . 4  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  <->  x  =  A ) )
23 velsn 3420 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
2422, 23syl6bbr 191 . . 3  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  <->  x  e.  { A } ) )
252, 24bitrd 181 . 2  |-  ( A  e.  RR*  ->  ( x  e.  ( A [,] A )  <->  x  e.  { A } ) )
2625eqrdv 2054 1  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   {csn 3403   class class class wbr 3792  (class class class)co 5540   RR*cxr 7118    < clt 7119    <_ cle 7120   [,]cicc 8861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltirr 7054  ax-pre-apti 7057
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-icc 8865
This theorem is referenced by: (None)
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