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Theorem ioodisj 9991
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 534 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  B  e.  RR* )
2 iooss1 9914 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  <_  C )  ->  ( C (,) D )  C_  ( B (,) D ) )
31, 2sylancom 420 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( C (,) D
)  C_  ( B (,) D ) )
4 ioossicc 9957 . . . . 5  |-  ( B (,) D )  C_  ( B [,] D )
53, 4sstrdi 3167 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( C (,) D
)  C_  ( B [,] D ) )
6 sslin 3361 . . . 4  |-  ( ( C (,) D ) 
C_  ( B [,] D )  ->  (
( A (,) B
)  i^i  ( C (,) D ) )  C_  ( ( A (,) B )  i^i  ( B [,] D ) ) )
75, 6syl 14 . . 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 533 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  A  e.  RR* )
9 simplrr 536 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  ->  D  e.  RR* )
10 df-ioo 9890 . . . . 5  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
11 df-icc 9893 . . . . 5  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
12 xrlenlt 8020 . . . . 5  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B  <_  w  <->  -.  w  <  B ) )
1310, 11, 12ixxdisj 9901 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  ->  (
( A (,) B
)  i^i  ( B [,] D ) )  =  (/) )
148, 1, 9, 13syl3anc 1238 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( B [,] D ) )  =  (/) )
157, 14sseqtrd 3193 . 2  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* ) )  /\  B  <_  C )  -> 
( ( A (,) B )  i^i  ( C (,) D ) ) 
C_  (/) )
16 ss0 3463 . 2  |-  ( ( ( A (,) B
)  i^i  ( C (,) D ) )  C_  (/) 
->  ( ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
1715, 16syl 14 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 104    = wceq 1353    e. wcel 2148    i^i cin 3128    C_ wss 3129   (/)c0 3422   class class class wbr 4003  (class class class)co 5874   RR*cxr 7989    < clt 7990    <_ cle 7991   (,)cioo 9886   [,]cicc 9889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-po 4296  df-iso 4297  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-ioo 9890  df-icc 9893
This theorem is referenced by: (None)
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