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Theorem ixxssixx 8872
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
ixxssixx.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, x    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( y,
z)

Proof of Theorem ixxssixx
StepHypRef Expression
1 ixxssixx.1 . . . 4  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
21elmpt2cl 5726 . . 3  |-  ( w  e.  ( A O B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
3 simp1 915 . . . . . 6  |-  ( ( w  e.  RR*  /\  A R w  /\  w S B )  ->  w  e.  RR* )
43a1i 9 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  w  e.  RR* ) )
5 simpl 106 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
6 3simpa 912 . . . . . 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 349 . . . . . 6  |-  ( A  e.  RR*  ->  ( ( w  e.  RR*  /\  A R w )  ->  A T w ) )
95, 6, 8syl2im 38 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  A T w ) )
10 simpr 107 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
11 3simpb 913 . . . . . 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 259 . . . . . . 7  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  (
w S B  ->  w U B ) )
1413expimpd 349 . . . . . 6  |-  ( B  e.  RR*  ->  ( ( w  e.  RR*  /\  w S B )  ->  w U B ) )
1510, 11, 14syl2im 38 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( w  e.  RR*  /\  A R w  /\  w S B )  ->  w U B ) )
164, 9, 153jcad 1096 . . . 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 8867 . . . 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 8867 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A P B )  <->  ( w  e.  RR*  /\  A T w  /\  w U B ) ) )
2016, 17, 193imtr4d 196 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  ->  w  e.  ( A P B ) ) )
212, 20mpcom 36 . 2  |-  ( w  e.  ( A O B )  ->  w  e.  ( A P B ) )
2221ssriv 2977 1  |-  ( A O B )  C_  ( A P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409   {crab 2327    C_ wss 2945   class class class wbr 3792  (class class class)co 5540    |-> cmpt2 5542   RR*cxr 7118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123
This theorem is referenced by:  ioossicc  8929  icossicc  8930  iocssicc  8931  ioossico  8932
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