Theorem ixxss2 9688

Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (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 ) } )
ixxss2.2	|- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z T y ) } )
ixxss2.3	|- ( ( w e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( w T B /\ B W C ) -> w S C ) )

Assertion

Ref	Expression
ixxss2	|- ( ( C e. RR* /\ B W C ) -> ( A P B ) C_ ( A O C ) )

Distinct variable groups:   x, w, y, z, A   w, C, 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   w, W

Allowed substitution hints:   P( x, y, z)   R( w)   S( w)   T( w)   O( y, z)   W( x, y, z)

Proof of Theorem ixxss2

Step	Hyp	Ref	Expression
1		ixxss2.2	. . . . . . . 8 |- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z T y ) } )
2	1	elixx3g 9684	. . . . . . 7 |- ( w e. ( A P B ) <-> ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) /\ ( A R w /\ w T B ) ) )
3	2	simplbi 272	. . . . . 6 |- ( w e. ( A P B ) -> ( A e. RR* /\ B e. RR* /\ w e. RR* ) )
4	3	adantl 275	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( A e. RR* /\ B e. RR* /\ w e. RR* ) )
5	4	simp3d 995	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w e. RR* )
6	2	simprbi 273	. . . . . 6 |- ( w e. ( A P B ) -> ( A R w /\ w T B ) )
7	6	adantl 275	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( A R w /\ w T B ) )
8	7	simpld 111	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> A R w )
9	7	simprd 113	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w T B )
10		simplr 519	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> B W C )
11	4	simp2d 994	. . . . . 6 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> B e. RR* )
12		simpll 518	. . . . . 6 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> C e. RR* )
13		ixxss2.3	. . . . . 6 |- ( ( w e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( w T B /\ B W C ) -> w S C ) )
14	5, 11, 12, 13	syl3anc 1216	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( ( w T B /\ B W C ) -> w S C ) )
15	9, 10, 14	mp2and 429	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w S C )
16	4	simp1d 993	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> A e. RR* )
17		ixxssixx.1	. . . . . 6 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
18	17	elixx1 9680	. . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( w e. ( A O C ) <-> ( w e. RR* /\ A R w /\ w S C ) ) )
19	16, 12, 18	syl2anc 408	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( w e. ( A O C ) <-> ( w e. RR* /\ A R w /\ w S C ) ) )
20	5, 8, 15, 19	mpbir3and 1164	. . 3 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w e. ( A O C ) )
21	20	ex 114	. 2 |- ( ( C e. RR* /\ B W C ) -> ( w e. ( A P B ) -> w e. ( A O C ) ) )
22	21	ssrdv 3103	1 |- ( ( C e. RR* /\ B W C ) -> ( A P B ) C_ ( A O C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   {crab 2420   C_ wss 3071   class class class wbr 3929  (class class class)co 5774   e. cmpo 5776   RR*cxr 7799

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804

This theorem is referenced by:  iooss2  9700