Theorem ixxss1 9875

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 9872	. . . . . . 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 1011	. . . 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 1009	. . . . . 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 1238	. . . . 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 1010	. . . . 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 9868	. . . . 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 1180	. . 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 3159	1 |- ( ( A e. RR* /\ A W B ) -> ( B P C ) C_ ( A O C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2146   {crab 2457   C_ wss 3127   class class class wbr 3998  (class class class)co 5865   e. cmpo 5867   RR*cxr 7965

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970

This theorem is referenced by:  iooss1  9887