MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixxdisj Structured version   Unicode version

Theorem ixxdisj 11547
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 3673 . . . 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 11541 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
433adant3 1014 . . . . . . . . 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 482 . . . . . . . 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 1008 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( A O B ) )  ->  w S B )
76adantrr 714 . . . . . 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 11541 . . . . . . . . . . 11  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( B P C )  <->  ( w  e.  RR*  /\  B T w  /\  w U C ) ) )
1093adant1 1012 . . . . . . . . . 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 482 . . . . . . . . 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 1007 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  B T w )
13 simpl2 998 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  B  e.  RR* )
1411simp1d 1006 . . . . . . . . 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 659 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  ( B T w  <->  -.  w S B ) )
1712, 16mpbid 210 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  w  e.  ( B P C ) )  ->  -.  w S B )
1817adantrl 713 . . . . . 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 574 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -.  ( w  e.  ( A O B )  /\  w  e.  ( B P C ) ) )
2019pm2.21d 106 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w  e.  ( A O B )  /\  w  e.  ( B P C ) )  ->  w  e.  (/) ) )
211, 20syl5bi 217 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
w  e.  ( ( A O B )  i^i  ( B P C ) )  ->  w  e.  (/) ) )
2221ssrdv 3495 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A O B )  i^i  ( B P C ) ) 
C_  (/) )
23 ss0 3815 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808    i^i cin 3460    C_ wss 3461   (/)c0 3783   class class class wbr 4439  (class class class)co 6270    |-> cmpt2 6272   RR*cxr 9616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-xr 9621
This theorem is referenced by:  ioodisj  11653  lecldbas  19887  icopnfcld  21441  iocmnfcld  21442  ioombl  22141  ismbf3d  22227  joiniooico  27819  asindmre  30342  dvasin  30343
  Copyright terms: Public domain W3C validator