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Theorem icodisj 8961
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 3154 . . . 4  |-  ( x  e.  ( ( A [,) B )  i^i  ( B [,) C
) )  <->  ( x  e.  ( A [,) B
)  /\  x  e.  ( B [,) C ) ) )
2 elico1 8893 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
323adant3 935 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
43biimpa 284 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( A [,) B
) )  ->  (
x  e.  RR*  /\  A  <_  x  /\  x  < 
B ) )
54simp3d 929 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( A [,) B
) )  ->  x  <  B )
65adantrr 456 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )  ->  x  <  B
)
7 elico1 8893 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
873adant1 933 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
98biimpa 284 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  (
x  e.  RR*  /\  B  <_  x  /\  x  < 
C ) )
109simp2d 928 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  <_  x )
11 simpl2 919 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  e.  RR* )
129simp1d 927 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  x  e.  RR* )
13 xrlenlt 7143 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  x  e.  RR* )  ->  ( B  <_  x  <->  -.  x  <  B ) )
1411, 12, 13syl2anc 397 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  ( B  <_  x  <->  -.  x  <  B ) )
1510, 14mpbid 139 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  -.  x  <  B )
1615adantrl 455 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )  ->  -.  x  <  B )
176, 16pm2.65da 597 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  ( x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )
1817pm2.21d 559 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) )  ->  x  e.  (/) ) )
191, 18syl5bi 145 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( ( A [,) B )  i^i  ( B [,) C ) )  ->  x  e.  (/) ) )
2019ssrdv 2979 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  C_  (/) )
21 ss0 3285 . 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 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409    i^i cin 2944    C_ wss 2945   (/)c0 3252   class class class wbr 3792  (class class class)co 5540   RR*cxr 7118    < clt 7119    <_ cle 7120   [,)cico 8860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-le 7125  df-ico 8864
This theorem is referenced by: (None)
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