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Theorem ixxdisj 11675
Description: Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
ixxun.2  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z U y ) } )
ixxun.3  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B T w  <->  -.  w S B ) )
Assertion
Ref Expression
ixxdisj  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A O B )  i^i  ( B P C ) )  =  (/) )
Distinct variable groups:    x, w, y, z, A    w, C, x, y, z    w, O   
w, B, x, y, z    w, P    x, R, y, z    x, S, y, z    x, T, y, z    x, U, y, z
Allowed substitution hints:    P( x, y, z)    R( w)    S( w)    T( w)    U( w)    O( x, y, z)

Proof of Theorem ixxdisj
StepHypRef Expression
1 elin 3608 . . . 4  |-  ( w  e.  ( ( A O B )  i^i  ( B P C ) )  <->  ( w  e.  ( A O B )  /\  w  e.  ( B P C ) ) )
2 ixx.1 . . . . . . . . . . 11  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
32elixx1 11669 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
433adant3 1050 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
54biimpa 492 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( A O B ) )  ->  (
w  e.  RR*  /\  A R w  /\  w S B ) )
65simp3d 1044 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( A O B ) )  ->  w S B )
76adantrr 731 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
w  e.  ( A O B )  /\  w  e.  ( B P C ) ) )  ->  w S B )
8 ixxun.2 . . . . . . . . . . . 12  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z U y ) } )
98elixx1 11669 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( B P C )  <->  ( w  e.  RR*  /\  B T w  /\  w U C ) ) )
1093adant1 1048 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
w  e.  ( B P C )  <->  ( w  e.  RR*  /\  B T w  /\  w U C ) ) )
1110biimpa 492 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  (
w  e.  RR*  /\  B T w  /\  w U C ) )
1211simp2d 1043 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  B T w )
13 simpl2 1034 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  B  e.  RR* )
1411simp1d 1042 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  w  e.  RR* )
15 ixxun.3 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  ( B T w  <->  -.  w S B ) )
1613, 14, 15syl2anc 673 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  ( B T w  <->  -.  w S B ) )
1712, 16mpbid 215 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  -.  w S B )
1817adantrl 730 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  (
w  e.  ( A O B )  /\  w  e.  ( B P C ) ) )  ->  -.  w S B )
197, 18pm2.65da 586 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  ( w  e.  ( A O B )  /\  w  e.  ( B P C ) ) )
2019pm2.21d 109 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w  e.  ( A O B )  /\  w  e.  ( B P C ) )  ->  w  e.  (/) ) )
211, 20syl5bi 225 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
w  e.  ( ( A O B )  i^i  ( B P C ) )  ->  w  e.  (/) ) )
2221ssrdv 3424 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A O B )  i^i  ( B P C ) ) 
C_  (/) )
23 ss0 3768 . 2  |-  ( ( ( A O B )  i^i  ( B P C ) ) 
C_  (/)  ->  ( ( A O B )  i^i  ( B P C ) )  =  (/) )
2422, 23syl 17 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A O B )  i^i  ( B P C ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {crab 2760    i^i cin 3389    C_ wss 3390   (/)c0 3722   class class class wbr 4395  (class class class)co 6308    |-> cmpt2 6310   RR*cxr 9692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-xr 9697
This theorem is referenced by:  ioodisj  11788  lecldbas  20312  icopnfcld  21866  iocmnfcld  21867  ioombl  22597  ismbf3d  22689  joiniooico  28431  asindmre  32091  dvasin  32092
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