Theorem ixxss12 9842

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

Step	Hyp	Ref	Expression
1		ixxss12.2	. . . . . . . 8 |- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z U y ) } )
2	1	elixx3g 9837	. . . . . . 7 |- ( w e. ( C P D ) <-> ( ( C e. RR* /\ D e. RR* /\ w e. RR* ) /\ ( C T w /\ w U D ) ) )
3	2	simplbi 272	. . . . . 6 |- ( w e. ( C P D ) -> ( C e. RR* /\ D e. RR* /\ w e. RR* ) )
4	3	adantl 275	. . . . 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* ) )
5	4	simp3d 1001	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> w e. RR* )
6		simplrl 525	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> A W C )
7	2	simprbi 273	. . . . . . 7 |- ( w e. ( C P D ) -> ( C T w /\ w U D ) )
8	7	adantl 275	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> ( C T w /\ w U D ) )
9	8	simpld 111	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> C T w )
10		simplll 523	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> A e. RR* )
11	4	simp1d 999	. . . . . 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 ) )
13	10, 11, 5, 12	syl3anc 1228	. . . . 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 ) )
14	6, 9, 13	mp2and 430	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> A R w )
15	8	simprd 113	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> w U D )
16		simplrr 526	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> D X B )
17	4	simp2d 1000	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> D e. RR* )
18		simpllr 524	. . . . . 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 ) )
20	5, 17, 18, 19	syl3anc 1228	. . . . 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 ) )
21	15, 16, 20	mp2and 430	. . . 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 ) } )
23	22	elixx1 9833	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
24	23	ad2antrr 480	. . . 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 ) ) )
25	5, 14, 21, 24	mpbir3and 1170	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) /\ w e. ( C P D ) ) -> w e. ( A O B ) )
26	25	ex 114	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) -> ( w e. ( C P D ) -> w e. ( A O B ) ) )
27	26	ssrdv 3148	1 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A W C /\ D X B ) ) -> ( C P D ) C_ ( A O B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   {crab 2448   C_ wss 3116   class class class wbr 3982  (class class class)co 5842   e. cmpo 5844   RR*cxr 7932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937

This theorem is referenced by:  iccss  9877  iccssioo  9878  icossico  9879  iccss2  9880  iccssico  9881  iocssioo  9899  icossioo  9900  ioossioo  9901