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Theorem ixxss2 10860
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
ixx.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   
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( x, 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 10854 . . . . . . 7  |-  ( w  e.  ( A P B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  /\  ( A R w  /\  w T B ) ) )
32simplbi 447 . . . . . 6  |-  ( w  e.  ( A P B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* ) )
43adantl 453 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( A  e.  RR*  /\  B  e. 
RR*  /\  w  e.  RR* ) )
54simp3d 971 . . . 4  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w  e.  RR* )
62simprbi 451 . . . . . 6  |-  ( w  e.  ( A P B )  ->  ( A R w  /\  w T B ) )
76adantl 453 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( A R w  /\  w T B ) )
87simpld 446 . . . 4  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  A R w )
97simprd 450 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w T B )
10 simplr 732 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  B W C )
114simp2d 970 . . . . . 6  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  B  e.  RR* )
12 simpll 731 . . . . . 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 1184 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( (
w T B  /\  B W C )  ->  w S C ) )
159, 10, 14mp2and 661 . . . 4  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w S C )
164simp1d 969 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  A  e.  RR* )
17 ixx.1 . . . . . 6  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
1817elixx1 10850 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A O C )  <->  ( w  e.  RR*  /\  A R w  /\  w S C ) ) )
1916, 12, 18syl2anc 643 . . . 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 1137 . . 3  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w  e.  ( A O C ) )
2120ex 424 . 2  |-  ( ( C  e.  RR*  /\  B W C )  ->  (
w  e.  ( A P B )  ->  w  e.  ( A O C ) ) )
2221ssrdv 3290 1  |-  ( ( C  e.  RR*  /\  B W C )  ->  ( A P B )  C_  ( A O C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2646    C_ wss 3256   class class class wbr 4146  (class class class)co 6013    e. cmpt2 6015   RR*cxr 9045
This theorem is referenced by:  iooss2  10877  leordtval2  17191  mnfnei  17200  psercnlem2  20200  tanord1  20299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-xr 9050
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