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Theorem snunioo 11649
Description: The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
snunioo  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )

Proof of Theorem snunioo
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 994 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  e.  RR* )
2 iccid 11577 . . . 4  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
31, 2syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( A [,] A )  =  { A } )
43uneq1d 3643 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A [,] A
)  u.  ( A (,) B ) )  =  ( { A }  u.  ( A (,) B ) ) )
5 simp2 995 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  RR* )
6 xrleid 11359 . . . 4  |-  ( A  e.  RR*  ->  A  <_  A )
71, 6syl 16 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <_  A )
8 simp3 996 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
9 df-icc 11539 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
10 df-ioo 11536 . . . 4  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
11 xrltnle 9642 . . . 4  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
12 df-ico 11538 . . . 4  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
13 xrlelttr 11362 . . . 4  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  A  /\  A  <  B )  ->  w  <  B
) )
14 xrltle 11358 . . . . . 6  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A  <_  w ) )
15143adant1 1012 . . . . 5  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  ( A  <  w  ->  A  <_  w ) )
1615adantld 465 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  A  /\  A  <  w )  ->  A  <_  w
) )
179, 10, 11, 12, 13, 16ixxun 11548 . . 3  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  A  /\  A  <  B ) )  -> 
( ( A [,] A )  u.  ( A (,) B ) )  =  ( A [,) B ) )
181, 1, 5, 7, 8, 17syl32anc 1234 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A [,] A
)  u.  ( A (,) B ) )  =  ( A [,) B ) )
194, 18eqtr3d 2497 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( { A }  u.  ( A (,) B ) )  =  ( A [,) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823    u. cun 3459   {csn 4016   class class class wbr 4439  (class class class)co 6270   RR*cxr 9616    < clt 9617    <_ cle 9618   (,)cioo 11532   [,)cico 11534   [,]cicc 11535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioo 11536  df-ico 11538  df-icc 11539
This theorem is referenced by:  prunioo  11652  ioojoin  11654  icombl1  22139  ioombl  22141  tan2h  30287  mbfposadd  30302  itg2addnclem2  30307  ftc1anclem5  30334  iocunico  31419  limciccioolb  31866  fourierdlem32  32160  fourierdlem93  32221
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