Theorem ixxss1 9528

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 ) } )
ixxss1.2	|- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z S y ) } )
ixxss1.3	|- ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) -> ( ( A W B /\ B T w ) -> A R w ) )

Assertion

Ref	Expression
ixxss1	|- ( ( A e. RR* /\ A W B ) -> ( B P C ) 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 ixxss1

Step	Hyp	Ref	Expression
1		ixxss1.2	. . . . . . . 8 |- P = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x T z /\ z S y ) } )
2	1	elixx3g 9525	. . . . . . 7 |- ( w e. ( B P C ) <-> ( ( B e. RR* /\ C e. RR* /\ w e. RR* ) /\ ( B T w /\ w S C ) ) )
3	2	simplbi 270	. . . . . 6 |- ( w e. ( B P C ) -> ( B e. RR* /\ C e. RR* /\ w e. RR* ) )
4	3	adantl 273	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( B e. RR* /\ C e. RR* /\ w e. RR* ) )
5	4	simp3d 963	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w e. RR* )
6		simplr 500	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A W B )
7	2	simprbi 271	. . . . . . 7 |- ( w e. ( B P C ) -> ( B T w /\ w S C ) )
8	7	adantl 273	. . . . . 6 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( B T w /\ w S C ) )
9	8	simpld 111	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> B T w )
10		simpll 499	. . . . . 6 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A e. RR* )
11	4	simp1d 961	. . . . . 6 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> B e. RR* )
12		ixxss1.3	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ w e. RR* ) -> ( ( A W B /\ B T w ) -> A R w ) )
13	10, 11, 5, 12	syl3anc 1184	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( ( A W B /\ B T w ) -> A R w ) )
14	6, 9, 13	mp2and 427	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A R w )
15	8	simprd 113	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w S C )
16	4	simp2d 962	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> C 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 9521	. . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( w e. ( A O C ) <-> ( w e. RR* /\ A R w /\ w S C ) ) )
19	10, 16, 18	syl2anc 406	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( w e. ( A O C ) <-> ( w e. RR* /\ A R w /\ w S C ) ) )
20	5, 14, 15, 19	mpbir3and 1132	. . 3 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w e. ( A O C ) )
21	20	ex 114	. 2 |- ( ( A e. RR* /\ A W B ) -> ( w e. ( B P C ) -> w e. ( A O C ) ) )
22	21	ssrdv 3053	1 |- ( ( A e. RR* /\ A W B ) -> ( B P C ) C_ ( A O C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 930   = wceq 1299   e. wcel 1448   {crab 2379   C_ wss 3021   class class class wbr 3875  (class class class)co 5706   e. cmpo 5708   RR*cxr 7671

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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676

This theorem is referenced by:  iooss1  9540