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Theorem joiniooico 28192
Description: Disjoint joining an open interval with a closed-below, open-above interval to form a closed-below, open-above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
Assertion
Ref Expression
joiniooico  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( ( ( A (,) B )  i^i  ( B [,) C
) )  =  (/)  /\  ( ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) ) )

Proof of Theorem joiniooico
Dummy variables  a 
b  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioo 11639 . . . 4  |-  (,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <  x  /\  x  <  b ) } )
2 df-ico 11641 . . . 4  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <_  x  /\  x  <  b ) } )
3 xrlenlt 9698 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
41, 2, 3ixxdisj 11650 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A (,) B
)  i^i  ( B [,) C ) )  =  (/) )
54adantr 466 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( ( A (,) B )  i^i  ( B [,) C ) )  =  (/) )
6 xrltletr 11454 . . 3  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w  <  B  /\  B  <_  C )  ->  w  <  C
) )
7 xrltletr 11454 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  w )  ->  A  <  w
) )
81, 2, 3, 1, 6, 7ixxun 11651 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) )
95, 8jca 534 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( ( ( A (,) B )  i^i  ( B [,) C
) )  =  (/)  /\  ( ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    u. cun 3440    i^i cin 3441   (/)c0 3767   class class class wbr 4426  (class class class)co 6305   RR*cxr 9673    < clt 9674    <_ cle 9675   (,)cioo 11635   [,)cico 11637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-ioo 11639  df-ico 11641
This theorem is referenced by:  difioo  28200
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