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Theorem ioounsn 36094
Description: The closure of the upper end of an open real interval. (Contributed by Jon Pennant, 8-Jun-2019.)
Assertion
Ref Expression
ioounsn  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )

Proof of Theorem ioounsn
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iccid 11681 . . . 4  |-  ( B  e.  RR*  ->  ( B [,] B )  =  { B } )
213ad2ant2 1030 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( B [,] B )  =  { B } )
32uneq2d 3588 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  ( B [,] B ) )  =  ( ( A (,) B )  u. 
{ B } ) )
4 simpl 459 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
5 simpr 463 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
64, 5, 53jca 1188 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* ) )
763adant3 1028 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* ) )
8 simp3 1010 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
9 xrleid 11449 . . . 4  |-  ( B  e.  RR*  ->  B  <_  B )
1093ad2ant2 1030 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  <_  B )
11 df-ioo 11639 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
12 df-icc 11642 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
13 xrlenlt 9699 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
14 df-ioc 11640 . . . 4  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
15 xrltle 11448 . . . . . 6  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w  <_  B ) )
16153adant3 1028 . . . . 5  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
w  <  B  ->  w  <_  B ) )
1716adantrd 470 . . . 4  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <  B  /\  B  <_  B )  ->  w  <_  B
) )
18 xrltletr 11454 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  w )  ->  A  <  w
) )
1911, 12, 13, 14, 17, 18ixxun 11651 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  B  /\  B  <_  B ) )  -> 
( ( A (,) B )  u.  ( B [,] B ) )  =  ( A (,] B ) )
207, 8, 10, 19syl12anc 1266 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  ( B [,] B ) )  =  ( A (,] B ) )
213, 20eqtr3d 2487 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    u. cun 3402   {csn 3968   class class class wbr 4402  (class class class)co 6290   RR*cxr 9674    < clt 9675    <_ cle 9676   (,)cioo 11635   (,]cioc 11636   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-pre-lttri 9613  ax-pre-lttrn 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-ioo 11639  df-ioc 11640  df-icc 11642
This theorem is referenced by:  iocunico  36095  iocmbl  36097
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