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Theorem joiniooico 27350
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 11534 . . . 4  |-  (,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <  x  /\  x  <  b ) } )
2 df-ico 11536 . . . 4  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <_  x  /\  x  <  b ) } )
3 xrlenlt 9653 . . . 4  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
41, 2, 3ixxdisj 11545 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A (,) B
)  i^i  ( B [,) C ) )  =  (/) )
54adantr 465 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( ( A (,) B )  i^i  ( B [,) C ) )  =  (/) )
6 xrltletr 11361 . . 3  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w  <  B  /\  B  <_  C )  ->  w  <  C
) )
7 xrltletr 11361 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <  B  /\  B  <_  w )  ->  A  <  w
) )
81, 2, 3, 1, 6, 7ixxun 11546 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  -> 
( ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) )
95, 8jca 532 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    u. cun 3474    i^i cin 3475   (/)c0 3785   class class class wbr 4447  (class class class)co 6285   RR*cxr 9628    < clt 9629    <_ cle 9630   (,)cioo 11530   [,)cico 11532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-pre-lttri 9567  ax-pre-lttrn 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-ioo 11534  df-ico 11536
This theorem is referenced by:  difioo  27358
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