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Theorem snunioo 10957
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 957 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  RR* )
2 iccid 10895 . . . 4  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
31, 2syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( A [,] A )  =  { A } )
43uneq1d 3445 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A [,] A
)  u.  ( A (,) B ) )  =  ( { A }  u.  ( A (,) B ) ) )
5 simp2 958 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  RR* )
6 xrleid 10677 . . . 4  |-  ( A  e.  RR*  ->  A  <_  A )
71, 6syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <_  A )
8 simp3 959 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
9 df-icc 10857 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
10 df-ioo 10854 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
11 xrltnle 9079 . . . 4  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
12 df-ico 10856 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
13 xrlelttr 10680 . . . 4  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  A  /\  A  <  B )  ->  w  <  B
) )
14 xrltle 10676 . . . . . 6  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A  <_  w ) )
15143adant1 975 . . . . 5  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  ( A  <  w  ->  A  <_  w ) )
1615adantld 454 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  A  /\  A  <  w )  ->  A  <_  w
) )
179, 10, 11, 12, 13, 16ixxun 10866 . . 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 1192 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A [,] A
)  u.  ( A (,) B ) )  =  ( A [,) B ) )
194, 18eqtr3d 2423 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    u. cun 3263   {csn 3759   class class class wbr 4155  (class class class)co 6022   RR*cxr 9054    < clt 9055    <_ cle 9056   (,)cioo 10850   [,)cico 10852   [,]cicc 10853
This theorem is referenced by:  prunioo  10959  ioojoin  10961  icombl1  19326  ioombl  19328  itg2addnclem2  25960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-pre-lttri 8999  ax-pre-lttrn 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-ioo 10854  df-ico 10856  df-icc 10857
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