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Theorem ioodisj 10765
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 735 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  B  e.  RR* )
2 iooss1 10691 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  <_  C )  ->  ( C (,) D )  C_  ( B (,) D ) )
31, 2sylancom 648 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( C (,) D
)  C_  ( B (,) D ) )
4 ioossicc 10735 . . . . 5  |-  ( B (,) D )  C_  ( B [,] D )
53, 4syl6ss 3191 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( C (,) D
)  C_  ( B [,] D ) )
6 sslin 3395 . . . 4  |-  ( ( C (,) D ) 
C_  ( B [,] D )  ->  (
( A (,) B
)  i^i  ( C (,) D ) )  C_  ( ( A (,) B )  i^i  ( B [,] D ) ) )
75, 6syl 15 . . 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 734 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  A  e.  RR* )
9 simplrr 737 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  D  e.  RR* )
10 df-ioo 10660 . . . . 5  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
11 df-icc 10663 . . . . 5  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
12 xrlenlt 8890 . . . . 5  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
1310, 11, 12ixxdisj 10671 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  ->  (
( A (,) B
)  i^i  ( B [,] D ) )  =  (/) )
148, 1, 9, 13syl3anc 1182 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( B [,] D ) )  =  (/) )
157, 14sseqtrd 3214 . 2  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) ) 
C_  (/) )
16 ss0 3485 . 2  |-  ( ( ( A (,) B
)  i^i  ( C (,) D ) )  C_  (/) 
->  ( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
1715, 16syl 15 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 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   (/)c0 3455   class class class wbr 4023  (class class class)co 5858   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656   [,]cicc 10659
This theorem is referenced by:  reconnlem1  18331  dyaddisjlem  18950  itgsplitioo  19192  ioodisjOLD  26230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-ioo 10660  df-icc 10663
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