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Theorem ixxss1 8874
Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
ixxssixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
ixxss1.2  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z S y ) } )
ixxss1.3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W B  /\  B T w )  ->  A R w ) )
Assertion
Ref Expression
ixxss1  |-  ( ( A  e.  RR*  /\  A W B )  ->  ( B P C )  C_  ( A O C ) )
Distinct variable groups:    x, w, y, z, A    w, C, x, y, z    w, O, x    w, B, x, y, z    w, P   
x, R, y, z   
x, S, y, z   
x, T, y, z   
w, W
Allowed substitution hints:    P( x, y, z)    R( w)    S( w)    T( w)    O( y, z)    W( x, y, z)

Proof of Theorem ixxss1
StepHypRef Expression
1 ixxss1.2 . . . . . . . 8  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z S y ) } )
21elixx3g 8871 . . . . . . 7  |-  ( w  e.  ( B P C )  <->  ( ( B  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  /\  ( B T w  /\  w S C ) ) )
32simplbi 263 . . . . . 6  |-  ( w  e.  ( B P C )  ->  ( B  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* ) )
43adantl 266 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( B  e.  RR*  /\  C  e. 
RR*  /\  w  e.  RR* ) )
54simp3d 929 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w  e.  RR* )
6 simplr 490 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A W B )
72simprbi 264 . . . . . . 7  |-  ( w  e.  ( B P C )  ->  ( B T w  /\  w S C ) )
87adantl 266 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( B T w  /\  w S C ) )
98simpld 109 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  B T w )
10 simpll 489 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A  e.  RR* )
114simp1d 927 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  B  e.  RR* )
12 ixxss1.3 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W B  /\  B T w )  ->  A R w ) )
1310, 11, 5, 12syl3anc 1146 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( ( A W B  /\  B T w )  ->  A R w ) )
146, 9, 13mp2and 417 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A R w )
158simprd 111 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w S C )
164simp2d 928 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  C  e.  RR* )
17 ixxssixx.1 . . . . . 6  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
1817elixx1 8867 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A O C )  <->  ( w  e.  RR*  /\  A R w  /\  w S C ) ) )
1910, 16, 18syl2anc 397 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( w  e.  ( A O C )  <->  ( w  e. 
RR*  /\  A R w  /\  w S C ) ) )
205, 14, 15, 19mpbir3and 1098 . . 3  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w  e.  ( A O C ) )
2120ex 112 . 2  |-  ( ( A  e.  RR*  /\  A W B )  ->  (
w  e.  ( B P C )  ->  w  e.  ( A O C ) ) )
2221ssrdv 2979 1  |-  ( ( A  e.  RR*  /\  A W B )  ->  ( B P C )  C_  ( A O C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ 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:  iooss1  8886
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