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Theorem snunioc 11411
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 11343 . . . 4  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
213ad2ant1 1009 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A [,] A )  =  { A } )
32uneq1d 3507 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,] A
)  u.  ( A (,] B ) )  =  ( { A }  u.  ( A (,] B ) ) )
4 simp1 988 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  RR* )
5 simp2 989 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  RR* )
6 xrleid 11125 . . . 4  |-  ( A  e.  RR*  ->  A  <_  A )
763ad2ant1 1009 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  A )
8 simp3 990 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  B )
9 df-icc 11305 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
10 df-ioc 11303 . . . 4  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
11 xrltnle 9441 . . . 4  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
12 xrletr 11130 . . . 4  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  (
( w  <_  A  /\  A  <_  B )  ->  w  <_  B
) )
13 simprr 756 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  <  w )
14 simpl1 991 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  e.  RR* )
15 simpl3 993 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  w  e.  RR* )
16 xrltle 11124 . . . . . . 7  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A  <_  w ) )
1714, 15, 16syl2anc 661 . . . . . 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 434 . . . 4  |-  ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  A  /\  A  <  w )  ->  A  <_  w
) )
209, 10, 11, 9, 12, 19ixxun 11314 . . 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 1226 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,] A
)  u.  ( A (,] B ) )  =  ( A [,] B ) )
223, 21eqtr3d 2475 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    u. cun 3324   {csn 3875   class class class wbr 4290  (class class class)co 6089   RR*cxr 9415    < clt 9416    <_ cle 9417   (,]cioc 11299   [,]cicc 11301
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-pre-lttri 9354  ax-pre-lttrn 9355
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-ioc 11303  df-icc 11305
This theorem is referenced by:  xrge0iifcnv  26361  xrge0iifiso  26363  xrge0iifhom  26365
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