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Theorem snunico 11024
Description: The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
snunico  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,) B
)  u.  { B } )  =  ( A [,] B ) )

Proof of Theorem snunico
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  RR* )
2 iccid 10961 . . . 4  |-  ( B  e.  RR*  ->  ( B [,] B )  =  { B } )
31, 2syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( B [,] B )  =  { B } )
43uneq2d 3501 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,) B
)  u.  ( B [,] B ) )  =  ( ( A [,) B )  u. 
{ B } ) )
5 simp1 957 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  RR* )
6 simp3 959 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  B )
7 xrleid 10743 . . . 4  |-  ( B  e.  RR*  ->  B  <_  B )
81, 7syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  <_  B )
9 df-ico 10922 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
10 df-icc 10923 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
11 xrlenlt 9143 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
12 xrltle 10742 . . . . . 6  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w  <_  B ) )
13123adant3 977 . . . . 5  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
w  <  B  ->  w  <_  B ) )
1413adantrd 455 . . . 4  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <  B  /\  B  <_  B )  ->  w  <_  B
) )
15 xrletr 10748 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  w )  ->  A  <_  w
) )
169, 10, 11, 10, 14, 15ixxun 10932 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  B  /\  B  <_  B ) )  -> 
( ( A [,) B )  u.  ( B [,] B ) )  =  ( A [,] B ) )
175, 1, 1, 6, 8, 16syl32anc 1192 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,) B
)  u.  ( B [,] B ) )  =  ( A [,] B ) )
184, 17eqtr3d 2470 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,) B
)  u.  { B } )  =  ( A [,] B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    u. cun 3318   {csn 3814   class class class wbr 4212  (class class class)co 6081   RR*cxr 9119    < clt 9120    <_ cle 9121   [,)cico 10918   [,]cicc 10919
This theorem is referenced by:  prunioo  11025  iccpnfcnv  18969  iccpnfhmeo  18970  xrge0iifcnv  24319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-ico 10922  df-icc 10923
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