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Theorem icodisj 11014
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 3522 . . . 4  |-  ( x  e.  ( ( A [,) B )  i^i  ( B [,) C
) )  <->  ( x  e.  ( A [,) B
)  /\  x  e.  ( B [,) C ) ) )
2 elico1 10951 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
323adant3 977 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
43biimpa 471 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( A [,) B
) )  ->  (
x  e.  RR*  /\  A  <_  x  /\  x  < 
B ) )
54simp3d 971 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( A [,) B
) )  ->  x  <  B )
65adantrr 698 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )  ->  x  <  B
)
7 elico1 10951 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
873adant1 975 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
98biimpa 471 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  (
x  e.  RR*  /\  B  <_  x  /\  x  < 
C ) )
109simp2d 970 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  <_  x )
11 simpl2 961 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  e.  RR* )
129simp1d 969 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  x  e.  RR* )
13 xrlenlt 9135 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  x  e.  RR* )  ->  ( B  <_  x  <->  -.  x  <  B ) )
1411, 12, 13syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  ( B  <_  x  <->  -.  x  <  B ) )
1510, 14mpbid 202 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  -.  x  <  B )
1615adantrl 697 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )  ->  -.  x  <  B )
176, 16pm2.65da 560 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  ( x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )
1817pm2.21d 100 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) )  ->  x  e.  (/) ) )
191, 18syl5bi 209 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( ( A [,) B )  i^i  ( B [,) C ) )  ->  x  e.  (/) ) )
2019ssrdv 3346 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  C_  (/) )
21 ss0 3650 . 2  |-  ( ( ( A [,) B
)  i^i  ( B [,) C ) )  C_  (/) 
->  ( ( A [,) B )  i^i  ( B [,) C ) )  =  (/) )
2220, 21syl 16 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204  (class class class)co 6073   RR*cxr 9111    < clt 9112    <_ cle 9113   [,)cico 10910
This theorem is referenced by:  icombl  19450  difico  24138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-xr 9116  df-le 9118  df-ico 10914
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