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Theorem ixxss2 11557
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 11551 . . . . . . 7  |-  ( w  e.  ( A P B )  <->  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  /\  ( A R w  /\  w T B ) ) )
32simplbi 460 . . . . . 6  |-  ( w  e.  ( A P B )  ->  ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* ) )
43adantl 466 . . . . 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 464 . . . . . 6  |-  ( w  e.  ( A P B )  ->  ( A R w  /\  w T B ) )
76adantl 466 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( A R w  /\  w T B ) )
87simpld 459 . . . 4  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  A R w )
97simprd 463 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  w T B )
10 simplr 755 . . . . 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 753 . . . . . 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 1229 . . . . 5  |-  ( ( ( C  e.  RR*  /\  B W C )  /\  w  e.  ( A P B ) )  ->  ( (
w T B  /\  B W C )  ->  w S C ) )
159, 10, 14mp2and 679 . . . 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 ixx.1 . . . . . 6  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
1817elixx1 11547 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A O C )  <->  ( w  e.  RR*  /\  A R w  /\  w S C ) ) )
1916, 12, 18syl2anc 661 . . . 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 434 . 2  |-  ( ( C  e.  RR*  /\  B W C )  ->  (
w  e.  ( A P B )  ->  w  e.  ( A O C ) ) )
2221ssrdv 3495 1  |-  ( ( C  e.  RR*  /\  B W C )  ->  ( A P B )  C_  ( A O C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   {crab 2797    C_ wss 3461   class class class wbr 4437  (class class class)co 6281    |-> cmpt2 6283   RR*cxr 9630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-xr 9635
This theorem is referenced by:  iooss2  11574  leordtval2  19586  mnfnei  19595  psercnlem2  22691  tanord1  22796
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