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Theorem ixxss12 10676
Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (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 ) } )
ixxss12.2  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z U y ) } )
ixxss12.3  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W C  /\  C T w )  ->  A R w ) )
ixxss12.4  |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e. 
RR* )  ->  (
( w U D  /\  D X B )  ->  w S B ) )
Assertion
Ref Expression
ixxss12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  ( C P D )  C_  ( A O B ) )
Distinct variable groups:    x, w, y, z, A    w, C, x, y, z    w, D, x, y, z    w, O    w, B, x, y, z    w, P    x, R, y, z    x, S, y, z    x, T, y, z    x, U, y, z    w, W   
w, X
Allowed substitution hints:    P( x, y, z)    R( w)    S( w)    T( w)    U( w)    O( x, y, z)    W( x, y, z)    X( x, y, z)

Proof of Theorem ixxss12
StepHypRef Expression
1 ixxss12.2 . . . . . . . 8  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z U y ) } )
21elixx3g 10669 . . . . . . 7  |-  ( w  e.  ( C P D )  <->  ( ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* )  /\  ( C T w  /\  w U D ) ) )
32simplbi 446 . . . . . 6  |-  ( w  e.  ( C P D )  ->  ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* ) )
43adantl 452 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  -> 
( C  e.  RR*  /\  D  e.  RR*  /\  w  e.  RR* ) )
54simp3d 969 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  RR* )
6 simplrl 736 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A W C )
72simprbi 450 . . . . . . 7  |-  ( w  e.  ( C P D )  ->  ( C T w  /\  w U D ) )
87adantl 452 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  -> 
( C T w  /\  w U D ) )
98simpld 445 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  C T w )
10 simplll 734 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A  e.  RR* )
114simp1d 967 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  C  e.  RR* )
12 ixxss12.3 . . . . . 6  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W C  /\  C T w )  ->  A R w ) )
1310, 11, 5, 12syl3anc 1182 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  -> 
( ( A W C  /\  C T w )  ->  A R w ) )
146, 9, 13mp2and 660 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A R w )
158simprd 449 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w U D )
16 simplrr 737 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D X B )
174simp2d 968 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D  e.  RR* )
18 simpllr 735 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  B  e.  RR* )
19 ixxss12.4 . . . . . 6  |-  ( ( w  e.  RR*  /\  D  e.  RR*  /\  B  e. 
RR* )  ->  (
( w U D  /\  D X B )  ->  w S B ) )
205, 17, 18, 19syl3anc 1182 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  -> 
( ( w U D  /\  D X B )  ->  w S B ) )
2115, 16, 20mp2and 660 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w S B )
22 ixx.1 . . . . . 6  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
2322elixx1 10665 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
2423ad2antrr 706 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  -> 
( w  e.  ( A O B )  <-> 
( w  e.  RR*  /\  A R w  /\  w S B ) ) )
255, 14, 21, 24mpbir3and 1135 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  ( A O B ) )
2625ex 423 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  (
w  e.  ( C P D )  ->  w  e.  ( A O B ) ) )
2726ssrdv 3185 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  ( C P D )  C_  ( A O B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   class class class wbr 4023  (class class class)co 5858    e. cmpt2 5860   RR*cxr 8866
This theorem is referenced by:  iccss  10718  iccssioo  10719  iccss2  10720  iccssico  10721  iocssioo  23261  icossioo  23262  icossico  23263  ioossioo  23264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-xr 8871
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