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Theorem snunioo1 37650
Description: The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
snunioo1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { A } )  =  ( A [,) B ) )

Proof of Theorem snunioo1
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uncom 3589 . 2  |-  ( ( A (,) B )  u.  ( A [,] A ) )  =  ( ( A [,] A )  u.  ( A (,) B ) )
2 iccid 11709 . . . 4  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
323ad2ant1 1035 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( A [,] A )  =  { A } )
43uneq2d 3599 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  ( A [,] A ) )  =  ( ( A (,) B )  u. 
{ A } ) )
5 simp1 1014 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  RR* )
6 simp2 1015 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  RR* )
7 xrleid 11477 . . . 4  |-  ( A  e.  RR*  ->  A  <_  A )
873ad2ant1 1035 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <_  A )
9 simp3 1016 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
10 df-icc 11670 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
11 df-ioo 11667 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
12 xrltnle 9726 . . . 4  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
13 df-ico 11669 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
14 xrlelttr 11481 . . . 4  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  A  /\  A  <  B )  ->  w  <  B
) )
15 simpl1 1017 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  e.  RR* )
16 simpl3 1019 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  w  e.  RR* )
17 simprr 771 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  <  w )
1815, 16, 17xrltled 37521 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  <_  w )
1918ex 440 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  A  /\  A  <  w )  ->  A  <_  w
) )
2010, 11, 12, 13, 14, 19ixxun 11679 . . 3  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  A  /\  A  <  B ) )  -> 
( ( A [,] A )  u.  ( A (,) B ) )  =  ( A [,) B ) )
215, 5, 6, 8, 9, 20syl32anc 1284 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A [,] A
)  u.  ( A (,) B ) )  =  ( A [,) B ) )
221, 4, 213eqtr3a 2519 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { A } )  =  ( A [,) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    u. cun 3413   {csn 3979   class class class wbr 4415  (class class class)co 6314   RR*cxr 9699    < clt 9700    <_ cle 9701   (,)cioo 11663   [,)cico 11665   [,]cicc 11666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-pre-lttri 9638  ax-pre-lttrn 9639
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-po 4773  df-so 4774  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-ioo 11667  df-ico 11669  df-icc 11670
This theorem is referenced by:  limcresioolb  37761  icocncflimc  37804  fourierdlem48  38055  fouriersw  38132  volico  38400
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