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Theorem ioounsn 35522
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 11545 . . . 4  |-  ( B  e.  RR*  ->  ( B [,] B )  =  { B } )
213ad2ant2 1019 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( B [,] B )  =  { B } )
32uneq2d 3596 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  ( B [,] B ) )  =  ( ( A (,) B )  u. 
{ B } ) )
4 simpl 455 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
5 simpr 459 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
64, 5, 53jca 1177 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* ) )
763adant3 1017 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* ) )
8 simp3 999 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
9 xrleid 11327 . . . 4  |-  ( B  e.  RR*  ->  B  <_  B )
1093ad2ant2 1019 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  <_  B )
11 df-ioo 11504 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
12 df-icc 11507 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
13 xrlenlt 9602 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
14 df-ioc 11505 . . . 4  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
15 xrltle 11326 . . . . . 6  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w  <_  B ) )
16153adant3 1017 . . . . 5  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
w  <  B  ->  w  <_  B ) )
1716adantrd 466 . . . 4  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <  B  /\  B  <_  B )  ->  w  <_  B
) )
18 xrltletr 11331 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  w )  ->  A  <  w
) )
1911, 12, 13, 14, 17, 18ixxun 11516 . . 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 1228 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  ( B [,] B ) )  =  ( A (,] B ) )
213, 20eqtr3d 2445 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    u. cun 3411   {csn 3971   class class class wbr 4394  (class class class)co 6234   RR*cxr 9577    < clt 9578    <_ cle 9579   (,)cioo 11500   (,]cioc 11501   [,]cicc 11503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-pre-lttri 9516  ax-pre-lttrn 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-ioo 11504  df-ioc 11505  df-icc 11507
This theorem is referenced by:  iocunico  35523  iocmbl  35525
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