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Theorem icodisj 10113
Description: End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
icodisj  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  =  (/) )

Proof of Theorem icodisj
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3355 . . . 4  |-  ( x  e.  ( ( A [,) B )  i^i  ( B [,) C
) )  <->  ( x  e.  ( A [,) B
)  /\  x  e.  ( B [,) C ) ) )
2 elico1 10044 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
323adant3 1019 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
43biimpa 296 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( A [,) B
) )  ->  (
x  e.  RR*  /\  A  <_  x  /\  x  < 
B ) )
54simp3d 1013 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( A [,) B
) )  ->  x  <  B )
65adantrr 479 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )  ->  x  <  B
)
7 elico1 10044 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
873adant1 1017 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
98biimpa 296 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  (
x  e.  RR*  /\  B  <_  x  /\  x  < 
C ) )
109simp2d 1012 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  <_  x )
11 simpl2 1003 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  e.  RR* )
129simp1d 1011 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  x  e.  RR* )
13 xrlenlt 8136 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  x  e.  RR* )  ->  ( B  <_  x  <->  -.  x  <  B ) )
1411, 12, 13syl2anc 411 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  ( B  <_  x  <->  -.  x  <  B ) )
1510, 14mpbid 147 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  -.  x  <  B )
1615adantrl 478 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )  ->  -.  x  <  B )
176, 16pm2.65da 662 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  ( x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )
1817pm2.21d 620 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) )  ->  x  e.  (/) ) )
191, 18biimtrid 152 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( ( A [,) B )  i^i  ( B [,) C ) )  ->  x  e.  (/) ) )
2019ssrdv 3198 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  C_  (/) )
21 ss0 3500 . 2  |-  ( ( ( A [,) B
)  i^i  ( B [,) C ) )  C_  (/) 
->  ( ( A [,) B )  i^i  ( B [,) C ) )  =  (/) )
2220, 21syl 14 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1372    e. wcel 2175    i^i cin 3164    C_ wss 3165   (/)c0 3459   class class class wbr 4043  (class class class)co 5943   RR*cxr 8105    < clt 8106    <_ cle 8107   [,)cico 10011
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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-xr 8110  df-le 8112  df-ico 10015
This theorem is referenced by: (None)
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