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Theorem ixxssixx 9320
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 5842 . . 3  |-  ( w  e.  ( A O B )  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
3 simp1 943 . . . . . 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 107 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
6 3simpa 940 . . . . . 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 355 . . . . . 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 108 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
11 3simpb 941 . . . . . 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 264 . . . . . . 7  |-  ( ( B  e.  RR*  /\  w  e.  RR* )  ->  (
w S B  ->  w U B ) )
1413expimpd 355 . . . . . 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 1124 . . . 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 9315 . . . 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 9315 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A P B )  <->  ( w  e.  RR*  /\  A T w  /\  w U B ) ) )
2016, 17, 193imtr4d 201 . . 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 3029 1  |-  ( A O B )  C_  ( A P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   {crab 2363    C_ wss 2999   class class class wbr 3845  (class class class)co 5652    |-> cmpt2 5654   RR*cxr 7521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-xr 7526
This theorem is referenced by:  ioossicc  9377  icossicc  9378  iocssicc  9379  ioossico  9380
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