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Theorem ioodisj 11639
Description: If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)
Assertion
Ref Expression
ioodisj  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )

Proof of Theorem ioodisj
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpllr 758 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  B  e.  RR* )
2 iooss1 11553 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  <_  C )  ->  ( C (,) D )  C_  ( B (,) D ) )
31, 2sylancom 667 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( C (,) D
)  C_  ( B (,) D ) )
4 ioossicc 11599 . . . . 5  |-  ( B (,) D )  C_  ( B [,] D )
53, 4syl6ss 3509 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( C (,) D
)  C_  ( B [,] D ) )
6 sslin 3717 . . . 4  |-  ( ( C (,) D ) 
C_  ( B [,] D )  ->  (
( A (,) B
)  i^i  ( C (,) D ) )  C_  ( ( A (,) B )  i^i  ( B [,] D ) ) )
75, 6syl 16 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) ) 
C_  ( ( A (,) B )  i^i  ( B [,] D
) ) )
8 simplll 757 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  A  e.  RR* )
9 simplrr 760 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  D  e.  RR* )
10 df-ioo 11522 . . . . 5  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
11 df-icc 11525 . . . . 5  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
12 xrlenlt 9641 . . . . 5  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
1310, 11, 12ixxdisj 11533 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  ->  (
( A (,) B
)  i^i  ( B [,] D ) )  =  (/) )
148, 1, 9, 13syl3anc 1223 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( B [,] D ) )  =  (/) )
157, 14sseqtrd 3533 . 2  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) ) 
C_  (/) )
16 ss0 3809 . 2  |-  ( ( ( A (,) B
)  i^i  ( C (,) D ) )  C_  (/) 
->  ( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
1715, 16syl 16 1  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    i^i cin 3468    C_ wss 3469   (/)c0 3778   class class class wbr 4440  (class class class)co 6275   RR*cxr 9616    < clt 9617    <_ cle 9618   (,)cioo 11518   [,]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-iun 4320  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-1st 6774  df-2nd 6775  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-ioo 11522  df-icc 11525
This theorem is referenced by:  reconnlem1  21059  dyaddisjlem  21732  itgsplitioo  21972
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