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Theorem ixxss12 10928
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 10921 . . . . . . 7  |-  ( w  e.  ( C P D )  <->  ( ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* )  /\  ( C T w  /\  w U D ) ) )
32simplbi 447 . . . . . 6  |-  ( w  e.  ( C P D )  ->  ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* ) )
43adantl 453 . . . . 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 971 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  RR* )
6 simplrl 737 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A W C )
72simprbi 451 . . . . . . 7  |-  ( w  e.  ( C P D )  ->  ( C T w  /\  w U D ) )
87adantl 453 . . . . . 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 446 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  C T w )
10 simplll 735 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A  e.  RR* )
114simp1d 969 . . . . . 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 1184 . . . . 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 661 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A R w )
158simprd 450 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w U D )
16 simplrr 738 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D X B )
174simp2d 970 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D  e.  RR* )
18 simpllr 736 . . . . . 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 1184 . . . . 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 661 . . . 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 10917 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
2423ad2antrr 707 . . . 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 1137 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  ( A O B ) )
2625ex 424 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  (
w  e.  ( C P D )  ->  w  e.  ( A O B ) ) )
2726ssrdv 3346 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 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:  iccss  10970  iccssioo  10971  icossico  10972  iccss2  10973  iccssico  10974  iocssioo  24124  icossioo  24125  ioossioo  24126  ftc1cnnclem  26268  ftc1anclem7  26276  ftc1anclem8  26277  ftc1anc  26278  ftc2nc  26279
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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