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Theorem ixxssixx 10670
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 6061 . . 3  |-  ( w  e.  ( A O B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
3 simp1 955 . . . . . 6  |-  ( ( w  e.  RR*  /\  A R w  /\  w S B )  ->  w  e.  RR* )
43a1i 10 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  w  e.  RR* ) )
5 simpl 443 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
6 3simpa 952 . . . . . 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 586 . . . . . 6  |-  ( A  e.  RR*  ->  ( ( w  e.  RR*  /\  A R w )  ->  A T w ) )
95, 6, 8syl2im 34 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  A T w ) )
10 simpr 447 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
11 3simpb 953 . . . . . 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 439 . . . . . . 7  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  (
w S B  ->  w U B ) )
1413expimpd 586 . . . . . 6  |-  ( B  e.  RR*  ->  ( ( w  e.  RR*  /\  w S B )  ->  w U B ) )
1510, 11, 14syl2im 34 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  w U B ) )
164, 9, 153jcad 1133 . . . 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 10665 . . . 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 10665 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A P B )  <->  ( w  e.  RR*  /\  A T w  /\  w U B ) ) )
2016, 17, 193imtr4d 259 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  ->  w  e.  ( A P B ) ) )
212, 20mpcom 32 . 2  |-  ( w  e.  ( A O B )  ->  w  e.  ( A P B ) )
2221ssriv 3184 1  |-  ( A O B )  C_  ( A P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   class class class wbr 4023  (class class class)co 5858    e. cmpt2 5860   RR*cxr 8866
This theorem is referenced by:  ioossicc  10735  itg2mulclem  19101  itg2mulc  19102  psercnlem2  19800  dvloglem  19995  icossicc  23258  iocssicc  23259  ioossico  23260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-xr 8871
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