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Theorem ixxss12 11424
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 11417 . . . . . . 7  |-  ( w  e.  ( C P D )  <->  ( ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* )  /\  ( C T w  /\  w U D ) ) )
32simplbi 460 . . . . . 6  |-  ( w  e.  ( C P D )  ->  ( C  e.  RR*  /\  D  e.  RR*  /\  w  e. 
RR* ) )
43adantl 466 . . . . 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 1002 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  RR* )
6 simplrl 759 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A W C )
72simprbi 464 . . . . . . 7  |-  ( w  e.  ( C P D )  ->  ( C T w  /\  w U D ) )
87adantl 466 . . . . . 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 459 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  C T w )
10 simplll 757 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A  e.  RR* )
114simp1d 1000 . . . . . 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 1219 . . . . 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 679 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  A R w )
158simprd 463 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w U D )
16 simplrr 760 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D X B )
174simp2d 1001 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  D  e.  RR* )
18 simpllr 758 . . . . . 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 1219 . . . . 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 679 . . . 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 11413 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
2423ad2antrr 725 . . . 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 1171 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  /\  w  e.  ( C P D ) )  ->  w  e.  ( A O B ) )
2625ex 434 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  (
w  e.  ( C P D )  ->  w  e.  ( A O B ) ) )
2726ssrdv 3463 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A W C  /\  D X B ) )  ->  ( C P D )  C_  ( A O B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2799    C_ wss 3429   class class class wbr 4393  (class class class)co 6193    |-> cmpt2 6195   RR*cxr 9521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-xr 9526
This theorem is referenced by:  iccss  11467  iccssioo  11468  icossico  11469  iccss2  11470  iccssico  11471  iocssioo  11489  icossioo  11490  ioossioo  11491  ftc1cnnclem  28606  ftc1anclem7  28614  ftc1anclem8  28615  ftc1anc  28616  ftc2nc  28617
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