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Theorem snunioo 11646
Description: The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
snunioo  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )

Proof of Theorem snunioo
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  RR* )
2 iccid 11574 . . . 4  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
31, 2syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( A [,] A )  =  { A } )
43uneq1d 3657 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A [,] A
)  u.  ( A (,) B ) )  =  ( { A }  u.  ( A (,) B ) ) )
5 simp2 997 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  RR* )
6 xrleid 11356 . . . 4  |-  ( A  e.  RR*  ->  A  <_  A )
71, 6syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <_  A )
8 simp3 998 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
9 df-icc 11536 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
10 df-ioo 11533 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
11 xrltnle 9653 . . . 4  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
12 df-ico 11535 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
13 xrlelttr 11359 . . . 4  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  A  /\  A  <  B )  ->  w  <  B
) )
14 xrltle 11355 . . . . . 6  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A  <_  w ) )
15143adant1 1014 . . . . 5  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  ( A  <  w  ->  A  <_  w ) )
1615adantld 467 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  A  /\  A  <  w )  ->  A  <_  w
) )
179, 10, 11, 12, 13, 16ixxun 11545 . . 3  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  A  /\  A  <  B ) )  -> 
( ( A [,] A )  u.  ( A (,) B ) )  =  ( A [,) B ) )
181, 1, 5, 7, 8, 17syl32anc 1236 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A [,] A
)  u.  ( A (,) B ) )  =  ( A [,) B ) )
194, 18eqtr3d 2510 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    u. cun 3474   {csn 4027   class class class wbr 4447  (class class class)co 6284   RR*cxr 9627    < clt 9628    <_ cle 9629   (,)cioo 11529   [,)cico 11531   [,]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-ioo 11533  df-ico 11535  df-icc 11536
This theorem is referenced by:  prunioo  11649  ioojoin  11651  icombl1  21736  ioombl  21738  tan2h  29652  mbfposadd  29667  itg2addnclem2  29672  ftc1anclem5  29699  iocunico  30811  limciccioolb  31191  cncfiooicclem1  31260  fourierdlem32  31467  fourierdlem93  31528
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