ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ixxss12 Unicode version

Theorem ixxss12 9385
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 9380 . . . . . . 7  |-  ( w  e.  ( C P D )  <->  ( ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* )  /\  ( C T w  /\  w U D ) ) )
32simplbi 269 . . . . . 6  |-  ( w  e.  ( C P D )  ->  ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* ) )
43adantl 272 . . . . 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 958 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  RR* )
6 simplrl 503 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A W C )
72simprbi 270 . . . . . . 7  |-  ( w  e.  ( C P D )  ->  ( C T w  /\  w U D ) )
87adantl 272 . . . . . 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 111 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  C T w )
10 simplll 501 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A  e.  RR* )
114simp1d 956 . . . . . 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 1175 . . . . 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 425 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A R w )
158simprd 113 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w U D )
16 simplrr 504 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D X B )
174simp2d 957 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D  e.  RR* )
18 simpllr 502 . . . . . 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 1175 . . . . 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 425 . . . 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 9376 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
2423ad2antrr 473 . . . 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 1127 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  ( A O B ) )
2625ex 114 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  (
w  e.  ( C P D )  ->  w  e.  ( A O B ) ) )
2726ssrdv 3032 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 103    <-> wb 104    /\ w3a 925    = wceq 1290    e. wcel 1439   {crab 2364    C_ wss 3000   class class class wbr 3851  (class class class)co 5666    |-> cmpt2 5668   RR*cxr 7582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-xr 7587
This theorem is referenced by:  iccss  9420  iccssioo  9421  icossico  9422  iccss2  9423  iccssico  9424  iocssioo  9442  icossioo  9443  ioossioo  9444
  Copyright terms: Public domain W3C validator