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Theorem ixxdisj 10931
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 3530 . . . 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 10925 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
433adant3 977 . . . . . . . . 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 471 . . . . . . . 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 971 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( A O B ) )  ->  w S B )
76adantrr 698 . . . . . 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 10925 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( B P C )  <->  ( w  e.  RR*  /\  B T w  /\  w U C ) ) )
1093adant1 975 . . . . . . . . . 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 471 . . . . . . . . 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 970 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  B T w )
13 simpl2 961 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  B  e.  RR* )
1411simp1d 969 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  ( B T w  <->  -.  w S B ) )
1712, 16mpbid 202 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  -.  w S B )
1817adantrl 697 . . . . . 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 560 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  ( w  e.  ( A O B )  /\  w  e.  ( B P C ) ) )
2019pm2.21d 100 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w  e.  ( A O B )  /\  w  e.  ( B P C ) )  ->  w  e.  (/) ) )
211, 20syl5bi 209 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
w  e.  ( ( A O B )  i^i  ( B P C ) )  ->  w  e.  (/) ) )
2221ssrdv 3354 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A O B )  i^i  ( B P C ) ) 
C_  (/) )
23 ss0 3658 . 2  |-  ( ( ( A O B )  i^i  ( B P C ) ) 
C_  (/)  ->  ( ( A O B )  i^i  ( B P C ) )  =  (/) )
2422, 23syl 16 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A O B )  i^i  ( B P C ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2709    i^i cin 3319    C_ wss 3320   (/)c0 3628   class class class wbr 4212  (class class class)co 6081    e. cmpt2 6083   RR*cxr 9119
This theorem is referenced by:  ioodisj  11026  lecldbas  17283  icopnfcld  18802  iocmnfcld  18803  ioombl  19459  ismbf3d  19546  joiniooico  24135
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-xr 9124
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