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Theorem ixxss2 10927
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 10921 . . . . . . 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 10917 . . . . 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 3346 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 1652    e. wcel 1725   {crab 2701    C_ wss 3312   class class class wbr 4204  (class class class)co 6073    e. cmpt2 6075   RR*cxr 9111
This theorem is referenced by:  iooss2  10944  leordtval2  17268  mnfnei  17277  psercnlem2  20332  tanord1  20431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-xr 9116
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