Theorem ixxss1 10100

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 10097	. . . . . . 7 |- ( w e. ( B P C ) <-> ( ( B e. RR* /\ C e. RR* /\ w e. RR* ) /\ ( B T w /\ w S C ) ) )
3	2	simplbi 274	. . . . . 6 |- ( w e. ( B P C ) -> ( B e. RR* /\ C e. RR* /\ w e. RR* ) )
4	3	adantl 277	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( B e. RR* /\ C e. RR* /\ w e. RR* ) )
5	4	simp3d 1035	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w e. RR* )
6		simplr 528	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A W B )
7	2	simprbi 275	. . . . . . 7 |- ( w e. ( B P C ) -> ( B T w /\ w S C ) )
8	7	adantl 277	. . . . . 6 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> ( B T w /\ w S C ) )
9	8	simpld 112	. . . . 5 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> B T w )
10		simpll 527	. . . . . 6 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A e. RR* )
11	4	simp1d 1033	. . . . . 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 1271	. . . . 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 433	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> A R w )
15	8	simprd 114	. . . 4 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w S C )
16	4	simp2d 1034	. . . . 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 10093	. . . . 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 411	. . . 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 1204	. . 3 |- ( ( ( A e. RR* /\ A W B ) /\ w e. ( B P C ) ) -> w e. ( A O C ) )
21	20	ex 115	. 2 |- ( ( A e. RR* /\ A W B ) -> ( w e. ( B P C ) -> w e. ( A O C ) ) )
22	21	ssrdv 3230	1 |- ( ( A e. RR* /\ A W B ) -> ( B P C ) C_ ( A O C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3197   class class class wbr 4083  (class class class)co 6001   e. cmpo 6003   RR*cxr 8180

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185

This theorem is referenced by:  iooss1  10112