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Theorem snunico 10810
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 956 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  RR* )
2 iccid 10748 . . . 4  |-  ( B  e.  RR*  ->  ( B [,] B )  =  { B } )
31, 2syl 15 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( B [,] B )  =  { B } )
43uneq2d 3363 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,) B
)  u.  ( B [,] B ) )  =  ( ( A [,) B )  u. 
{ B } ) )
5 simp1 955 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  RR* )
6 simp3 957 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  B )
7 xrleid 10531 . . . 4  |-  ( B  e.  RR*  ->  B  <_  B )
81, 7syl 15 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  <_  B )
9 df-ico 10709 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
10 df-icc 10710 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
11 xrlenlt 8935 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
12 xrltle 10530 . . . . . 6  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w  <_  B ) )
13123adant3 975 . . . . 5  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
w  <  B  ->  w  <_  B ) )
1413adantrd 454 . . . 4  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <  B  /\  B  <_  B )  ->  w  <_  B
) )
15 xrletr 10536 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  w )  ->  A  <_  w
) )
169, 10, 11, 10, 14, 15ixxun 10719 . . 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 1190 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,) B
)  u.  ( B [,] B ) )  =  ( A [,] B ) )
184, 17eqtr3d 2350 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 934    = wceq 1633    e. wcel 1701    u. cun 3184   {csn 3674   class class class wbr 4060  (class class class)co 5900   RR*cxr 8911    < clt 8912    <_ cle 8913   [,)cico 10705   [,]cicc 10706
This theorem is referenced by:  prunioo  10811  iccpnfcnv  18495  iccpnfhmeo  18496  xrge0iifcnv  23388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-pre-lttri 8856  ax-pre-lttrn 8857
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-ico 10709  df-icc 10710
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