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Theorem snunioc 11637
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 11563 . . . 4  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
213ad2ant1 1012 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A [,] A )  =  { A } )
32uneq1d 3650 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,] A
)  u.  ( A (,] B ) )  =  ( { A }  u.  ( A (,] B ) ) )
4 simp1 991 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  RR* )
5 simp2 992 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  RR* )
6 xrleid 11345 . . . 4  |-  ( A  e.  RR*  ->  A  <_  A )
763ad2ant1 1012 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  A )
8 simp3 993 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  B )
9 df-icc 11525 . . . 4  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
10 df-ioc 11523 . . . 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 xrletr 11350 . . . 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 994 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  A  e.  RR* )
15 simpl3 996 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  A  e.  RR*  /\  w  e.  RR* )  /\  ( A  <_  A  /\  A  <  w ) )  ->  w  e.  RR* )
16 xrltle 11344 . . . . . . 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 11534 . . 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 1231 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( A [,] A
)  u.  ( A (,] B ) )  =  ( A [,] B ) )
223, 21eqtr3d 2503 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 968    = wceq 1374    e. wcel 1762    u. cun 3467   {csn 4020   class class class wbr 4440  (class class class)co 6275   RR*cxr 9616    < clt 9617    <_ cle 9618   (,]cioc 11519   [,]cicc 11521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioc 11523  df-icc 11525
This theorem is referenced by:  xrge0iifcnv  27537  xrge0iifiso  27539  xrge0iifhom  27541
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