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

Theorem ixxssixx 10922
Description: An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
Hypotheses
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
ixx.2  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z U y ) } )
ixx.3  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A T w ) )
ixx.4  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w S B  ->  w U B ) )
Assertion
Ref Expression
ixxssixx  |-  ( A O B )  C_  ( A P B )
Distinct variable groups:    x, w, y, z, A    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 ixxssixx
StepHypRef Expression
1 ixx.1 . . . 4  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
21elmpt2cl 6280 . . 3  |-  ( w  e.  ( A O B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
3 simp1 957 . . . . . 6  |-  ( ( w  e.  RR*  /\  A R w  /\  w S B )  ->  w  e.  RR* )
43a1i 11 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  w  e.  RR* ) )
5 simpl 444 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
6 3simpa 954 . . . . . 6  |-  ( ( w  e.  RR*  /\  A R w  /\  w S B )  ->  (
w  e.  RR*  /\  A R w ) )
7 ixx.3 . . . . . . 7  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A T w ) )
87expimpd 587 . . . . . 6  |-  ( A  e.  RR*  ->  ( ( w  e.  RR*  /\  A R w )  ->  A T w ) )
95, 6, 8syl2im 36 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  A T w ) )
10 simpr 448 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
11 3simpb 955 . . . . . 6  |-  ( ( w  e.  RR*  /\  A R w  /\  w S B )  ->  (
w  e.  RR*  /\  w S B ) )
12 ixx.4 . . . . . . . 8  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w S B  ->  w U B ) )
1312ancoms 440 . . . . . . 7  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  (
w S B  ->  w U B ) )
1413expimpd 587 . . . . . 6  |-  ( B  e.  RR*  ->  ( ( w  e.  RR*  /\  w S B )  ->  w U B ) )
1510, 11, 14syl2im 36 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  w U B ) )
164, 9, 153jcad 1135 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  -> 
( w  e.  RR*  /\  A T w  /\  w U B ) ) )
171elixx1 10917 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
18 ixx.2 . . . . 5  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z U y ) } )
1918elixx1 10917 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A P B )  <->  ( w  e.  RR*  /\  A T w  /\  w U B ) ) )
2016, 17, 193imtr4d 260 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  ->  w  e.  ( A P B ) ) )
212, 20mpcom 34 . 2  |-  ( w  e.  ( A O B )  ->  w  e.  ( A P B ) )
2221ssriv 3344 1  |-  ( A O B )  C_  ( A P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   class class class wbr 4204  (class class class)co 6073    e. cmpt2 6075   RR*cxr 9111
This theorem is referenced by:  ioossicc  10988  itg2mulclem  19630  itg2mulc  19631  psercnlem2  20332  dvloglem  20531  icossicc  24121  iocssicc  24122  ioossico  24123
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-pow 4369  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
  Copyright terms: Public domain W3C validator