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Theorem snunioc 11702
Description: The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)
Assertion
Ref Expression
snunioc  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( { A }  u.  ( A (,] B ) )  =  ( A [,] B ) )

Proof of Theorem snunioc
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iccid 11627 . . . 4  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
213ad2ant1 1018 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A [,] A )  =  { A } )
32uneq1d 3596 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,] A
)  u.  ( A (,] B ) )  =  ( { A }  u.  ( A (,] B ) ) )
4 simp1 997 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  RR* )
5 simp2 998 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  RR* )
6 xrleid 11409 . . . 4  |-  ( A  e.  RR*  ->  A  <_  A )
763ad2ant1 1018 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  A )
8 simp3 999 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  B )
9 df-icc 11589 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
10 df-ioc 11587 . . . 4  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
11 xrltnle 9683 . . . 4  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
12 xrletr 11414 . . . 4  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  A  /\  A  <_  B )  ->  w  <_  B
) )
13 simprr 758 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  <  w )
14 simpl1 1000 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  e.  RR* )
15 simpl3 1002 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  w  e.  RR* )
16 xrltle 11408 . . . . . . 7  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A  <_  w ) )
1714, 15, 16syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  -> 
( A  <  w  ->  A  <_  w )
)
1813, 17mpd 15 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  <_  w )
1918ex 432 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  A  /\  A  <  w )  ->  A  <_  w
) )
209, 10, 11, 9, 12, 19ixxun 11598 . . 3  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  A  /\  A  <_  B ) )  -> 
( ( A [,] A )  u.  ( A (,] B ) )  =  ( A [,] B ) )
214, 4, 5, 7, 8, 20syl32anc 1238 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,] A
)  u.  ( A (,] B ) )  =  ( A [,] B ) )
223, 21eqtr3d 2445 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    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    u. cun 3412   {csn 3972   class class class wbr 4395  (class class class)co 6278   RR*cxr 9657    < clt 9658    <_ cle 9659   (,]cioc 11583   [,]cicc 11585
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 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-pre-lttri 9596  ax-pre-lttrn 9597
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 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-ioc 11587  df-icc 11589
This theorem is referenced by:  xrge0iifcnv  28368  xrge0iifiso  28370  xrge0iifhom  28372
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