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Theorem icodisj 11646
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 3687 . . . 4  |-  ( x  e.  ( ( A [,) B )  i^i  ( B [,) C
) )  <->  ( x  e.  ( A [,) B
)  /\  x  e.  ( B [,) C ) ) )
2 elico1 11573 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
323adant3 1016 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( A [,) B )  <->  ( x  e.  RR*  /\  A  <_  x  /\  x  <  B
) ) )
43biimpa 484 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( A [,) B
) )  ->  (
x  e.  RR*  /\  A  <_  x  /\  x  < 
B ) )
54simp3d 1010 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( A [,) B
) )  ->  x  <  B )
65adantrr 716 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )  ->  x  <  B
)
7 elico1 11573 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
873adant1 1014 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( B [,) C )  <->  ( x  e.  RR*  /\  B  <_  x  /\  x  <  C
) ) )
98biimpa 484 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  (
x  e.  RR*  /\  B  <_  x  /\  x  < 
C ) )
109simp2d 1009 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  <_  x )
11 simpl2 1000 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  B  e.  RR* )
129simp1d 1008 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  x  e.  RR* )
13 xrlenlt 9653 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  x  e.  RR* )  ->  ( B  <_  x  <->  -.  x  <  B ) )
1411, 12, 13syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  ( B  <_  x  <->  -.  x  <  B ) )
1510, 14mpbid 210 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  x  e.  ( B [,) C
) )  ->  -.  x  <  B )
1615adantrl 715 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )  ->  -.  x  <  B )
176, 16pm2.65da 576 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  ( x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) ) )
1817pm2.21d 106 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( x  e.  ( A [,) B )  /\  x  e.  ( B [,) C ) )  ->  x  e.  (/) ) )
191, 18syl5bi 217 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
x  e.  ( ( A [,) B )  i^i  ( B [,) C ) )  ->  x  e.  (/) ) )
2019ssrdv 3510 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A [,) B
)  i^i  ( B [,) C ) )  C_  (/) )
21 ss0 3816 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   (/)c0 3785   class class class wbr 4447  (class class class)co 6285   RR*cxr 9628    < clt 9629    <_ cle 9630   [,)cico 11532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-xr 9633  df-le 9635  df-ico 11536
This theorem is referenced by:  icombl  21801  difico  27359
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