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Theorem ixxss12 9005
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
ixxssixx.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, x    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( 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 9000 . . . . . . 7  |-  ( w  e.  ( C P D )  <->  ( ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* )  /\  ( C T w  /\  w U D ) ) )
32simplbi 268 . . . . . 6  |-  ( w  e.  ( C P D )  ->  ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* ) )
43adantl 271 . . . . 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 953 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  RR* )
6 simplrl 502 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A W C )
72simprbi 269 . . . . . . 7  |-  ( w  e.  ( C P D )  ->  ( C T w  /\  w U D ) )
87adantl 271 . . . . . 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 110 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  C T w )
10 simplll 500 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A  e.  RR* )
114simp1d 951 . . . . . 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 1170 . . . . 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 424 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A R w )
158simprd 112 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w U D )
16 simplrr 503 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D X B )
174simp2d 952 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D  e.  RR* )
18 simpllr 501 . . . . . 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 1170 . . . . 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 424 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w S B )
22 ixxssixx.1 . . . . . 6  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
2322elixx1 8996 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
2423ad2antrr 472 . . . 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 1122 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  ( A O B ) )
2625ex 113 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  (
w  e.  ( C P D )  ->  w  e.  ( A O B ) ) )
2726ssrdv 3006 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    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   {crab 2353    C_ wss 2974   class class class wbr 3793  (class class class)co 5543    |-> cmpt2 5545   RR*cxr 7214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219
This theorem is referenced by:  iccss  9040  iccssioo  9041  icossico  9042  iccss2  9043  iccssico  9044  iocssioo  9062  icossioo  9063  ioossioo  9064
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