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Theorem ixxss2 9903
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 ) } )
ixxss2.2  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z T y ) } )
ixxss2.3  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w T B  /\  B W C )  ->  w S C ) )
Assertion
Ref Expression
ixxss2  |-  ( ( C  e.  RR*  /\  B W C )  ->  ( A P B )  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 ixxss2
StepHypRef Expression
1 ixxss2.2 . . . . . . . 8  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z T y ) } )
21elixx3g 9899 . . . . . . 7  |-  ( w  e.  ( A P B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  /\  ( A R w  /\  w T B ) ) )
32simplbi 274 . . . . . 6  |-  ( w  e.  ( A P B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* ) )
43adantl 277 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( A  e.  RR*  /\  B  e. 
RR*  /\  w  e.  RR* ) )
54simp3d 1011 . . . 4  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w  e.  RR* )
62simprbi 275 . . . . . 6  |-  ( w  e.  ( A P B )  ->  ( A R w  /\  w T B ) )
76adantl 277 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( A R w  /\  w T B ) )
87simpld 112 . . . 4  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  A R w )
97simprd 114 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w T B )
10 simplr 528 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  B W C )
114simp2d 1010 . . . . . 6  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  B  e.  RR* )
12 simpll 527 . . . . . 6  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  C  e.  RR* )
13 ixxss2.3 . . . . . 6  |-  ( ( w  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( w T B  /\  B W C )  ->  w S C ) )
145, 11, 12, 13syl3anc 1238 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( (
w T B  /\  B W C )  ->  w S C ) )
159, 10, 14mp2and 433 . . . 4  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w S C )
164simp1d 1009 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  A  e.  RR* )
17 ixxssixx.1 . . . . . 6  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
1817elixx1 9895 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A O C )  <->  ( w  e.  RR*  /\  A R w  /\  w S C ) ) )
1916, 12, 18syl2anc 411 . . . 4  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( w  e.  ( A O C )  <->  ( w  e. 
RR*  /\  A R w  /\  w S C ) ) )
205, 8, 15, 19mpbir3and 1180 . . 3  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w  e.  ( A O C ) )
2120ex 115 . 2  |-  ( ( C  e.  RR*  /\  B W C )  ->  (
w  e.  ( A P B )  ->  w  e.  ( A O C ) ) )
2221ssrdv 3161 1  |-  ( ( C  e.  RR*  /\  B W C )  ->  ( A P B )  C_  ( A O C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   {crab 2459    C_ wss 3129   class class class wbr 4003  (class class class)co 5874    e. cmpo 5876   RR*cxr 7989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994
This theorem is referenced by:  iooss2  9915
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