MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iccid Structured version   Unicode version

Theorem iccid 11574
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 11573 . . . 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 9652 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <_  x  <->  -.  x  <  A ) )
4 xrlenlt 9652 . . . . . . . . . . 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 11349 . . . . . . . . . . . . 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 1210 . . . . 5  |-  ( A  e.  RR*  ->  ( ( x  e.  RR*  /\  A  <_  x  /\  x  <_  A )  ->  x  =  A ) )
15 eleq1a 2550 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  e.  RR* ) )
16 xrleid 11356 . . . . . . 7  |-  ( A  e.  RR*  ->  A  <_  A )
17 breq2 4451 . . . . . . 7  |-  ( x  =  A  ->  ( A  <_  x  <->  A  <_  A ) )
1816, 17syl5ibrcom 222 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  A  <_  x ) )
19 breq1 4450 . . . . . . 7  |-  ( x  =  A  ->  (
x  <_  A  <->  A  <_  A ) )
2016, 19syl5ibrcom 222 . . . . . 6  |-  ( A  e.  RR*  ->  ( x  =  A  ->  x  <_  A ) )
2115, 18, 203jcad 1177 . . . . 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 4041 . . . 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 2464 1  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {csn 4027   class class class wbr 4447  (class class class)co 6284   RR*cxr 9627    < clt 9628    <_ cle 9629   [,]cicc 11532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-icc 11536
This theorem is referenced by:  snunioo  11646  snunico  11647  snunioc  11648  prunioo  11649  icccmplem1  21090  ivthicc  21633  ioombl  21738  volivth  21779  mbfimasn  21804  itgspliticc  22006  dvivth  22174  cvmliftlem10  28407  mblfinlem2  29657  areacirc  29717  ioounsn  30810  iocinico  30812  iocmbl  30813  snunioo2  31136  snunioo1  31144  cncfiooicc  31261
  Copyright terms: Public domain W3C validator