Theorem ixxss2 10184

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 10180	. . . . . . 7 |- ( w e. ( A P B ) <-> ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) /\ ( A R w /\ w T B ) ) )
3	2	simplbi 274	. . . . . 6 |- ( w e. ( A P B ) -> ( A e. RR* /\ B e. RR* /\ w e. RR* ) )
4	3	adantl 277	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( A e. RR* /\ B e. RR* /\ w e. RR* ) )
5	4	simp3d 1038	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w e. RR* )
6	2	simprbi 275	. . . . . 6 |- ( w e. ( A P B ) -> ( A R w /\ w T B ) )
7	6	adantl 277	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> ( A R w /\ w T B ) )
8	7	simpld 112	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> A R w )
9	7	simprd 114	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w T B )
10		simplr 529	. . . . 5 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> B W C )
11	4	simp2d 1037	. . . . . 6 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> B e. RR* )
12		simpll 527	. . . . . 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 1274	. . . . 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 433	. . . 4 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w S C )
16	4	simp1d 1036	. . . . 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 10176	. . . . 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 411	. . . 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 1207	. . 3 |- ( ( ( C e. RR* /\ B W C ) /\ w e. ( A P B ) ) -> w e. ( A O C ) )
21	20	ex 115	. 2 |- ( ( C e. RR* /\ B W C ) -> ( w e. ( A P B ) -> w e. ( A O C ) ) )
22	21	ssrdv 3234	1 |- ( ( C e. RR* /\ B W C ) -> ( A P B ) C_ ( A O C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   {crab 2515   C_ wss 3201   class class class wbr 4093  (class class class)co 6028   e. cmpo 6030   RR*cxr 8255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260

This theorem is referenced by:  iooss2  10196