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Theorem ioodisj 11018
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 736 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  B  e.  RR* )
2 iooss1 10943 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  <_  C )  ->  ( C (,) D )  C_  ( B (,) D ) )
31, 2sylancom 649 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( C (,) D
)  C_  ( B (,) D ) )
4 ioossicc 10988 . . . . 5  |-  ( B (,) D )  C_  ( B [,] D )
53, 4syl6ss 3352 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( C (,) D
)  C_  ( B [,] D ) )
6 sslin 3559 . . . 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 735 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  A  e.  RR* )
9 simplrr 738 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  D  e.  RR* )
10 df-ioo 10912 . . . . 5  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
11 df-icc 10915 . . . . 5  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
12 xrlenlt 9135 . . . . 5  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
1310, 11, 12ixxdisj 10923 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  ->  (
( A (,) B
)  i^i  ( B [,] D ) )  =  (/) )
148, 1, 9, 13syl3anc 1184 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( B [,] D ) )  =  (/) )
157, 14sseqtrd 3376 . 2  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) ) 
C_  (/) )
16 ss0 3650 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204  (class class class)co 6073   RR*cxr 9111    < clt 9112    <_ cle 9113   (,)cioo 10908   [,]cicc 10911
This theorem is referenced by:  reconnlem1  18849  dyaddisjlem  19479  itgsplitioo  19721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-ioo 10912  df-icc 10915
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