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Theorem icodisj 10114
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 3356 . . . 4  |-  ( x  e.  ( ( A [,) B )  i^i  ( B [,) C
) )  <->  ( x  e.  ( A [,) B
)  /\  x  e.  ( B [,) C ) ) )
2 elico1 10045 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
323adant3 1020 . . . . . . . . 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 1014 . . . . . . 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 10045 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
873adant1 1018 . . . . . . . . . 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 1013 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  <_  x )
11 simpl2 1004 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  e.  RR* )
129simp1d 1012 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  x  e.  RR* )
13 xrlenlt 8137 . . . . . . . . 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 663 . . . . 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 3199 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  C_  (/) )
21 ss0 3501 . 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 981    = wceq 1373    e. wcel 2176    i^i cin 3165    C_ wss 3166   (/)c0 3460   class class class wbr 4044  (class class class)co 5944   RR*cxr 8106    < clt 8107    <_ cle 8108   [,)cico 10012
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-le 8113  df-ico 10016
This theorem is referenced by: (None)
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