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Theorem ioounsn 29585
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 11345 . . . 4  |-  ( B  e.  RR*  ->  ( B [,] B )  =  { B } )
213ad2ant2 1010 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( B [,] B )  =  { B } )
32uneq2d 3510 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  ( B [,] B ) )  =  ( ( A (,) B )  u. 
{ B } ) )
4 simpl 457 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
5 simpr 461 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
64, 5, 53jca 1168 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* ) )
763adant3 1008 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* ) )
8 simp3 990 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
9 xrleid 11127 . . . 4  |-  ( B  e.  RR*  ->  B  <_  B )
1093ad2ant2 1010 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  <_  B )
11 df-ioo 11304 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
12 df-icc 11307 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
13 xrlenlt 9442 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
14 df-ioc 11305 . . . 4  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
15 xrltle 11126 . . . . . 6  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w  <_  B ) )
16153adant3 1008 . . . . 5  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
w  <  B  ->  w  <_  B ) )
1716adantrd 468 . . . 4  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <  B  /\  B  <_  B )  ->  w  <_  B
) )
18 xrltletr 11131 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  w )  ->  A  <  w
) )
1911, 12, 13, 14, 17, 18ixxun 11316 . . 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 1216 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  ( B [,] B ) )  =  ( A (,] B ) )
213, 20eqtr3d 2477 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    u. cun 3326   {csn 3877   class class class wbr 4292  (class class class)co 6091   RR*cxr 9417    < clt 9418    <_ cle 9419   (,)cioo 11300   (,]cioc 11301   [,]cicc 11303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-pre-lttri 9356  ax-pre-lttrn 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-ioo 11304  df-ioc 11305  df-icc 11307
This theorem is referenced by:  iocunico  29586  iocmbl  29588
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